P O RO ELAS TI C STRUCTURES
P O RO ELAS TI C STRUCTURES
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P 0 RO ELAS TI C STRUCTURES Gabriel Cederbaum Department of Mechanical Engineering BenGurion University of the Negev BeerSheva, Israel
LePing Li Biomedical Engineering Institute Ecole Polytechnique of Montreal Montreal, Canada
Kalman Schulgasser Department of Mechanical Engineering BenGurion University of the Negev BeerSheva, Israel
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ELSEVIER
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To our beloved children
Jonathan and Daniel Cederbaum Hon g Yi Li Joshua, Yael, Daniel and Noam Schulgasser
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Contents
CHAPTER 1. Introduction
1
CHAPTER 2. Modeling of poroelastic beams 2.1. Basic equations 2.2. Characteristic times 2.3. Note on a beam impermeable at both ends 2.4. Equations in nondimensional form
9 9 15 16 17
CHAPTER 3. Analytical solutions for quasistatic beams 3.1. Simplysupported beams with permeable ends 3.2. Beams subjected to loads suddenly applied and constant thereafter
19 19 20
CHAPTER 4. Finite element formulation and solutions for quasistatic beams 4.1. Introduction 4.2. Variational principles 4.3. Finite element formulation 4.4. Examples and discussion
33 33 34 37 41
CHAPTER 5. Vibrations of poroelastic beams 5.1. Initial value problems 5.2. Forced harmonic vibrations 5.3. Closure
53 53 63 65
CHAPTER 6. Large deflection analysis of poreolastic beams 6.1. Governing equations 6.2. Equations in nondimensional form when F ~ = O 6.3. Numerical formulation 6.4. Numerical procedure for the finite difference method 6.5. Examples and discussion
67 67 71 72 77 79
...
\ 111
Contents
CHAPTER 7. Stability of poroelastic columns 7.1. Buckling of columns 7.2. Limits of critical load 7.3. Timedependence of critical load and deflections 7.4. Postbuckling: formulation 7.5. Postbuckling: results and discussion 7.6. Imperfection sensitivity 7.7. Dynamic stability of poroelastic columns 7.8. Stability boundaries and critical load amplitude
89 89 90 92 96 99 102 104 107
CHAPTER 8. Analysis of poroelastic plates 8.1. Basic equations for thin plates 8.2. Bending equations in nondimensional form 8.3. Analytical solutions for quasistatic bending 8.4. Transverse vibrations of simply supported plates
111 111 118 119 128
CHAPTER 9. Closure
135
REFERENCES
141
APPENDIX A. Proof of the variational principles
145
APPENDIX B. A finite element for poroelastic beams
149
APPENDIX C. Several useful Laplace inverse transformations
151
APPENDIX D. Coefficients in difference formulas
153
APPENDIX E. Determination of boundary values at x1 for the finite difference method
155
SUBJECT INDEX
157
Chapter 1 INTRODUCTION
Poroelasticity is a continuum theory for the analysis of a porous media consisting of an elastic matrix containing interconnected fluidsaturated pores. In physical terms the theory postulates that when a porous material is subjected to stress, the resulting matrix deformation leads to volumetric changes in the pores. Since the pores are fluidfilled, the presence of the fluid not only acts as a stiffener of the material, but also results in the flow of the pore fluid (diffusion) between regions of higher and lower pore pressure. If the fluid is viscous the behavior of the material system becomes time dependent. The basic phenomenological model for such a material was proposed by Biot (Biot, 1941a,b, 1955, 1956a,b, 1962, 1964; Biot and Willis, 1957). His motivation (and the application of the theory over the years) was concerned with soil consolidation (quasistatic) and wave propagation (dynamic) problems in geomechanics. The literature dealing with poroelasticity based on the now classical model is voluminous. It cannot possibly be reviewed within the scope of the present work. Most of this literature deals with very specific aspects of the theory or with specific physical problems. We cite a selection of general and fundamental references in which numerous other citations can be found: Bear (1972, 1990), Cheng et al. (1993), Cleary (1977), Coussy (1995), Detourney and Cheng (1993), Kumpel (1991), Rice and Cleary (1976), Rudnicki (1985) and Thomson and Willis (1991). We point out that simultaneous with the development of poroelasticity, literature dealing with thermoelasticity has evolved. Here it is assumed that there is a coupling between the thermal diffusion equations and the equations of mechanical equilibrium. The complete mathematical analogy of the poroelastic and thermoelastic problems was noted by Biot (1956c). A comprehensive treatment from this latter point of view is given by Nowacki (1986). A parallel literature has been produced, but it is far less voluminous, probably due to the lesser occurrence of instances when it is physically significant to consider coupling in the thermoelastic case. Poroelastic theories were originally motivated by problems in soil and geomechanics. This is the point of departure of most of the literature cited above. These problems generally concern massive structures and are by nature threedimensional. Consolidation problems, seismic wave propagation, crustal dynamics and seabed mechanics are some examples. This application of poroelastic theory is relatively mature. In the past two decades the poroelastic model has also been extensively and successfully applied in biomechanics. There are similarities in mechanical properties and behaviors between
2
Ch. 1
Introduction
geomechanical and certain biological structures. For example, it has been found that certain porous rocks, marbles and granites have material properties that are similar to those of bone (e.g. Cowin, 1999). The pores in such biological structures are interconnected so that the pore fluid can transport nutrients to, and take waste away from, the cells. One of the most important functions of the fluid in articular cartilage is to lubricate the joints and thus protect them from wearing. The mechanism of the fluid flow in bone and cartilage is so 'designed' that it also protects the biological structure from damage resulting from dynamic loading. Application of poroelastic theory in biomechanics is presently very vigorous. Therefore it is appropriate to review some of this work. This gives some indication of the potential fertility of this material model for other than the original application for which the idea was developed. Poroelastic models of bone were first reported 30 years ago (e.g. Nowinski and Davis, 1970, 1972; Nowinski, 1971, 1972; Johnson et al., 1982). The bulk modulus of the matrix of bone is usually several times higher than that of the fluid in the bone. Thus the pore fluid does not share much of the the mechanical loading when loaded slowly. However, this is not true in the case of dynamic loading  the transient stiffness can be much higher than the stiffness of the drained bone. The importance of such studies using poroelasticity in bone mechanics also lies in understanding the fluid motion, which induces the streaming potentials (electrical potentials produced by ionic motion) and transports nutrients. A survey of the application of poroelasticity in bone mechanics has been given by Cowin (1999). As opposed to bone, soft tissue has a much lower elastic modulus and thus the pore fluid carries a significant portion of the mechanical load. For articular cartilage, the transient stiffness may be ten times as high as the drained stiffness under physiological loadings. Poroelastic theories have been extensively employed for the study of cartilage and other soft tissue (e.g. Mow et al., 1984, 1993; Eisenberg and Grodzinsky, 1985, 1987). Uniaxial compression tests (e.g. Armstrong et al., 1984; Lanir, 1987), indentation tests (e.g. Mak et al., 1987; Spilker et al., 1992b), joint contact analyses (e.g. Ateshian and Wang, 1995; Wu et al., 1996), determination of streaming potentials (e.g. Gu et al., 1993, 1998), and other situations have been considered. Finite element methods have been generally used to extract solutions (e.g. Goldsmith et al., 1996; Simon, 1992; Spilker and Suh, 1990). Due to the complicated behavior of the biological fluids and the sophisticated matrix structure different models have been offered (e.g. Mow et al., 1980; Mak, 1986a,b; Lai et al., 1991; Setton et al., 1993; Almeida and Spilker, 1998; Suh and Bai, 1998; Li et al., 1999b, 2000; Suh and DiSilvestro, 1999). Other applications of poroelastic models include analyses of spinal motion segments (Simon et al., 1985a,b; Laible et al., 1993; Argoubi and ShiraziAdl, 1996), meniscus of knee joints (Spilker et al., 1992a), blood flow through soft tissue (Vankan et al., 1996, 1997) and orthopedic implants (e.g. Prendergast, 1997). Relatively few papers have thus far investigated the poroelastic beams or plates, the light structures, for which the boundary conditions and the type of loading, and thus the behavior of the structure, are quite different from those for large formations. When such elements are subjected to bending, the stress gradients would generally be expected to be much greater in the perpendicular direction than in the axial or inplane directions. Thus if the bulk material can be considered to be isotropic, the diffusion in the transverse direction is dominant. Hence, in the studies reported in the literature the diffusion in the axial or inplane directions is, justifiably, considered negligible and the fluid movement
Introduction
3
in the perpendicular direction has been taken as the prevailing diffusion effect. Among such papers available is that of Nowinski and Davis (1972), who modeled a beam as an anisotropic poroelastic body subjected to a uniform bending moment or a uniform torsion moment. This is a case where no diffusion occurs in the direction of the beam axis since the stress gradients are zero in that direction. Taber (1992) and Theodorakopoulos and Beskos (1993, 1994) formulated the Kirchhoff plate, assuming that the fluidvelocity gradients within the plate plane relative to the solid are negligible. Zhang and Cowin (1994) considered combinations of pure bending and axial compression for rectangular beams, again for no diffusion in the direction of the beam axis. Biot (1964) discussed the buckling problem for a plate with special deformation: in one direction within the plate, the normal strain is zero; in the perpendicular direction within the plate, the fluid displacement relative to the solid is zero. Thus the fluid can flow only in the transverse direction. This book is devoted to the analysis of fluidsaturated poroelastic beams, columns and plates made of materials for which diffusion in the longitudinal direction(s) is viable, while in the perpendicular direction(s) the flow can be considered negligible because of the microgeometry of the solid skeletal material. Our initial motivation to investigate such structures is mainly related to plant stems and petioles. These elements of plants serve the dual functions of providing structural strength and stiffness, and also contain the vascular tissue, which conducts water from the root system to transpiring leaves. Living herbaceous stems and woody stem tissue are water saturated, the former often containing as much as 85% free water by weight, the latter as much as 60%. Such structural plant material is highly anisotropic. The axial stiffness is some 20 times greater than transverse stiffness for woody tissue, and for other plant tissue the anisotropy is probably much greater (Schulgasser and Witztum, 1992). The crucial attribute of such plant elements is that their microstructure is designed to transport water axially. Plants native to nondesert areas transport enormous quantities of water daily from the root systems through stems which are transpired from the leaves (Weier et al., 1982). We are thus led to model a living plant stem as a beam consisting of a poroelastic material for which water movement in the axial direction is dominant. While the overwhelming bulk of literature on poroelasticity in general, and of the references already cited in particular, deals with media which are taken to be statistically isotropic, the theory has also been developed for the anisotropic case (e.g. Carroll, 1979; Rudnicki, 1985; Thomson and Willis, 1991). The underlying physical assumptions are the same. However, the increased mathematical complexity in obtaining solutions, the profusion of material constants whose values must be determined, and also the fact that for most applications previously considered, isotropy is a reasonable model, has resulted in the concentration on the isotropic case. Here we take the anisotropy to the extreme and consider elements whose dimension in the direction perpendicular to fluid motion is small compared to dimensions in the flow direction(s). On the one hand, no new physical assumptions are involved, and on the other hand, for this special case, the number of material parameters is eminently tractable. The material constituting the beam elements will be taken to be transversely isotropic in the crosssectional plane. The study provides a methodology and a theoretical basis for investigating the mechanical behaviors of the structural elements made of such materials. These are not limited to the materials of plant
4
Ch. 1
Introduction
stems, which was the original motivation for the material model; artificial materials with similar behavior are of greater interest and potential. Many microstructures and fabrication schemes could be imagined, which would produce bulk materials with the postulated behavior. It is not the purpose of this book to deal with micromechanics or with the relationship of microstructure to effective bulk properties. Nor is it our purpose to investigate appropriate fabrication procedures. We simply point out that such materials could be produced with a wide range of bulk behaviors, and could be tailored to specific situations. We only demonstrate how easily any closed pore foam (i.e. the pores are not interconnected) can be converted to the type of material under consideration. This is illustrated in Fig. 1.1. We have simply pierced the (previously closed pore) model with a battery of parallel needles. The behavior of light structures (beams, columns, plates) constituted of such material was first investigated by the present authors in a series of articles (Li et al., 1995, 1996, 1997a,b,c, 1999a; Cederbaum et al., 1998; Cederbaum, 2000a,b). See also Li (1997). The governing equations for a transversely isotropic poroelastic beam (transversely isotropic in the crosssection) subjected to transverse and/or axial loads, as obtained within the small deflection theory, are presented in Chapter 2, including the inertia of the bulk material. Biot's theory, with relative motion between the solid and fluid governed by Darcy's law, is adapted for the case considered. The governing differential equations can be separated into two groups, one for bending and another for extension; since they are not coupled, they can be solved independently. Each group includes three equations for three unknown timedependent functions: the total stress resultant, the pore pressure resultant and the displacement. The conditions for determination of solutions include the geometrical boundary conditions, the load boundary conditions and the diffusion boundary conditions, as well as the initial conditions. Chapter 3 presents analytical solutions for the quasistatic bending problem of beams. The formulation is derived by deleting the inertia term from the partial differential equation, which governs the equilibrium of the beam. The elastic solutions, i.e. the solutions for the corresponding drained beams, are introduced in order to simplify the solution procedure so that various closed form solutions for the poroelastic beams can be found. Series solutions are found for normal loading with various mechanical and diffusion boundary conditions. Due to the complexity of the boundaries and the governing differential equations, it is often difficult to get analytical solutions for general cases, especially when the boundary
Fig. 1.1. 'Handmade' poroelastic material with axial diffusion.
Introduction
5
conditions are not homogenous or when they cannot be decoupled. Therefore, finding suitable numerical methods for respective problems is an important part of the present work. The finite element method is employed for the quasistatic beams and columns under small deflection in Chapter 4. Variational principles are first developed. The variational functional is expressed in terms of integrals of the unknown timedependent functions with respect to position on the beam and convolution integrals with respect to time. Two types of variables, the displacements and pore pressure resultants, are involved in the timedependent functionals. The method of Lagrange multipliers is employed in order to include the flow equations (generalized Darcy's law) in the EulerLagrange equations of the functionals. Two functionals are given: one includes the initial values of the unknown functions and is more convenient for the interpolation of the displacement velocities; another functional is more convenient for the interpolation of the displacements. Both functionals are found to be equivalent to each other in terms of their stationary conditions, which give the governing differential equations and boundary conditions. A mixed finite element scheme is then presented based on one of the variational functionals obtained. Numerical solution examples for both types of variables are presented in order to test the finite element model, and good coincidence with the previously found analytical solutions is shown. The results also demonstrate some unique features of poroelastic beams and columns, which cannot be shown by examples for which analytical solutions can be found. In Chapter 5 solutions are found for free and forced vibration situations of the poroelastic beams. Closed form solutions of the initial value problems are obtained for simply supported beams with general loading by use of Laplace transformation. It turns out that the fluid works as a damper. Similar to the classic vibration theory of damped elastic beams, the responses to initial deviations can be classified into three types: light damping, critical damping and over damping. The vibration patterns are also dependent on the nature of the initial conditions; observed behavior of this sort cannot be explained by the classical vibration theory. Computations for forced harmonic vibrations are carried out for different boundaries. The amplitude response versus the frequency of the loading, and the resonance areas, are investigated. Chapter 6 deals with large deflections of beams. While in the previous chapters the deflection was considered to be small, and thus linear theories are sufficient when the constitutive law is linear, for some situations it may be necessary to employ a large deflection theory in order to correctly describe the behavior of the structure. On the other hand, the deformation can be still small and the skeletal material yet behaves elastically; the large deflection is made possible by the slenderness of the beam. Therefore, it is modeled as geometrically nonlinear and constitutively linear. Biot's constitutive law and Darcy's law are adopted as in the linear theory, while new geometrical relations and equilibrium equations are necessarily introduced. In the large deflection case the stretching and bending problems are coupled. The nonlinear boundary value problem is solved numerically by using the finite difference method with respect to the spatial coordinate and using a simple successive implicit formula (the trapezoid formula) to deal with the time variable. Several types of geometrical and diffusion boundary conditions are investigated by means of numerical solutions. Results are presented, for which observations are made, and some interesting features are found which do not occur when the problem is modeled as linear (i.e. small deflections).
6
Ch. 1 Introduction
The stability of poroelastic columns is investigated in Chapter 7. Three problems are considered: buckling, postbuckling and dynamic stability. For the buckling problem, the time dependent behaviors of the critical loads and deflections are considered for various diffusion and geometrical boundaries. Upper (short time) and lower (long time) limits for the timedependent critical load are found for the case of a timedependent load. It is also shown that buckling can be avoided during a loading procedure by properly choosing the loading path, even when the load at finite time is greater than the lower limit of the critical load. For the postbuckling problem, the timedependent behavior of the columns, governed by three coupled equations, is obtained by using the large deflection theory. These equations are transformed into a single one, enabling the analytical derivation of the initial and the final responses. It is shown that unlike the quasistatic response obtained by using the small deflection theory, the long time response derived here is bounded. The imperfection sensitivity of these columns is also investigated. For the dynamic stability problem, stability conditions and boundaries are derived. It is shown that the stability regions are expanded with respect to the elastic (drained) case. The critical (minimum) loading amplitude for which instability occurs is also given. Formulations are found in Chapter 8 for fluidsaturated poroelastic plates consisting of a material for which the diffusion is possible in the inplane directions only, both for bending and for inplane loading. The plates considered are isotropic in the plate plane, and the Kirchhoff hypotheses are assumed. Again Biot's constitutive law is adopted and Darcy's law is used to describe the fluid flow in pores. The basic equations are so derived that they could be easily extended for the situation of an orthotropic poroelastic plate. Closed form solutions are extracted for quasistatic problems and for vibrations. Observations are made on the types of deflection/vibration patterns, which are obtained. Finally in Chapter 9 we present a beam model composed of a discrete elastic structure containing a fluid which together will behave in an identical manner to the beam composed of a poroelastic continuum used in the formulation of Chapter 2. This discrete model has heuristic value in appreciating the phenomena. We show how adjusting the physical parameters of the materials and the geometric parameters influence the extent of the poroelastic effect. Throughout the book some very unique features of the proposed model are shown (in some cases a comparison is made with another timedependent model, viscoelasticity). First, the mechanical behaviors of the structural elements are shown to be greatly dependent on the diffusion patterns which are in turn dependent on the loading, on the geometrical and diffusion boundary conditions, as well as on the material parameters. This is in contrast to the case of problems of the same kind of structures (as far as geometry and loading) with diffusion in the transverse direction, where the transverse diffusion patterns for different positions along the beam are all similar. Second, three time scales are required to describe the vibration system, as compared to two in viscoelasticity. Moreover, the vibration patterns of the present system are determined not only by the parameters of the material and the geometry of the structure, but also by the initial pore pressure conditions. This implies that even in the case of 'light damping' oscillatory motion may not occur for some initial pore pressure conditions, which would not be explainable if the structures were modeled as viscoelastic. Third, the pore pressure at a given position does not necessarily decay monotonically after a suddenly applied and then constant loading; it may
Introduction
7
increase for some time and then decay toward the final value in some cases. This phenomenon is similar to the socalled MandelCryer effect. Moreover, the pore pressure at other positions can possibly increase monotonically for all time; in some cases the sign of pore pressure can even change twice during the diffusion process. Fourth, the pore fluid works also as a damper. This damping mechanism is useful in reducing vibrations. By changing the properties of the fluid, which for instance can result from a temperature increment, or by altering the diffusion boundaries, a resonance can be avoided or an oscillatory motion could be made to disappear when so desired. It is also possible to avoid oscillatory motions by choosing initial pore pressure conditions, without changing the properties of the material and the geometry of the structural element. Again, this is quite different from a viscoelastic damping system. As a result of the features mentioned above, it is recognized that the response of the poroelastic structural element to loading is sensitive to the properties of the fluid and to the diffusion boundaries, which can be easily altered in practice. Therefore, such structural elements and thus their features are potentially controllable. In other words, it could be possible to convert such elements into intelligent or smart structures. If this is so, it would be interesting that such structural elements could work as both sensors and actuators, e.g. the fluid can 'feel' the change of the temperature by changing its viscosity and this results in a change of the behavior of the structure. This book attempts to constitute a reasonably selfcontained presentation of a wide spectrum of problems related to the analysis of the type of poroelastic structure considered. It is hoped that the book will serve as an inspiration, guide, and reference for applying such elements in mechanical, biomechanical, civil and aerospace engineering, as well as a textbook for graduate studies.
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Chapter 2 MODELING OF POROELASTIC BEAMS
In this chapter formulations are established for poroelastic beams subjected to axial and transverse loads, when the fluid diffusion is axial by virtue of the material microgeometry. The material is taken to be transversely isotropic in the crosssectional plane. The basic assumptions employed in formulating the geometrical relations are those of classical beam theory. Following the basic formulations, general features of the system are demonstrated. Finally the nondimensional equations are obtained.
2.1. Basic Equations Consider the beam shown in Fig. 2.1. The area moment of inertia of the crosssection with respect to the yaxis is I; crosssectional area is A, which is taken to be symmetric with respect to the zaxis. The mass per unit length of the beam is p. The beam is subjected to a distributed normal load qn(X, t) and a distributed axial load qs(X, t). As noted above, the elastic porous matrix is saturated with a fluid, and the microstructure is such that the beam material is transversely isotropic and permits fluid movement in the axial direction only (or at least predominantly). The classical beam theory is suitable for a beam whose crosssectional dimensions are small compared to its length. The basic assumptions and implications of this theory are: 9 plane crosssections of the beam remain plane after deformation 9 upon the application of load the crosssection undergoes, at most, a translation and a rotation with respect to the original coordinate system 9 a normal to the crosssection remains normal to this surface 9 the rotational inertia is negligible, and 9 the only nonnegligible normal stress is the x component of normal stress. In the present situation this stress is averaged over both solid and fluid phases. Thus %c = Ox  ~bpf
(2.1)
where Ox is the partial stress acting on the solid skeleton, pf is the pore pressure in the fluid, and ~b is the porosity, i.e. the pore volume fraction. The internal axial force and moment shown in Fig. 2.1 are the resultants of the normal stress, and are thus given by
10
Ch. 2
N i
!
Modeling o f Poroelastic B e a m s
t
Fig. 2.1. Beam subjected to distributed normal and distributed axial loads, showing also the internal forces. The inertia moment of the crosssectionwith respect to the yaxis is I and the mass per unit length of the beam is p.
N  [ z "rx dA,
M  ~A "rxZ d A
(2.2a, b)
The equilibrium equations for the small deflection problem of an elastic beam are still valid for the poroelastic beam; these are ON 0x 
32u qs + P 0tT
oZM
OZw
o~X2
  qn + P O t 2
OM
0x
(2.3a)
(2.3b)
 Q
(2.3c)
where u and w are the axial and transverse displacements, respectively, of the beam axis (at z = 0) and Q is the shear force. It has been assumed in writing the last term on the fight of Eq. (2.3a) that the acceleration of the fluid relative to the solid skeleton is small. This is justified from a practical point of view by the low permeability to be expected in such materials. As a result no inertia terms reflecting the motion of the fluid with respect to the solid are included. The rotational inertia (of both the fluid and the solid) may also be neglected. The kinematic assumption above permits the writing of the following relationships for the horizontal displacement and strain at any point in the solid skeleton of the material: u s  u  ZW, x,
~~  U,x  ZW, xx
(2.4a, b)
where the subscript comma denotes xdirection derivatives, and W,x is the rotation of the crosssection. The constitutive equations for a transversely isotropic poroelastic material, as given by Biot (1962) and using his notation, are
Basic Equations
ry
=
rz
11
B3
B2 + 2B1
82
4
B3
B2
B 2 + 2B 1
8~
+
B6 ~
(2.5a)
B6 (2.5b)
p f  B7gSx t B6( 4 + 8~) + B8~
Here ~" is the increment of fluid content in the pores (or pore volume change); it is the porosity times the difference in traces of the strain in the solid and in the fluid, i.e. ~" = 4~(es  e e)
(2.6)
where es
eSx + @ + gsz
and
~f
efx + eyf + ,s'z f
(2.7a, b)
The superscripts s and f are used to refer to the solid and fluid phases, respectively. There are seven independent material constants in the constitutive equations; due to the third assumption of classical beam theory, those involving shear stress need not be and hence are not included. Eqs. (2.5) are rewritten in the form "1"x
'ry
I Cll  C12
C12 C22
C12
C23
"1z
pf
E. It is further noted that E and 77 are independent of the only possible relevant mechanical property of the fluid, i.e. its compressibility, while A depends both on the mechanical properties and microstructure of the skeleton, as well as on the fluid compressibility. When ~"= 0 we further record, for later use, another relationship which is valid, A P f    ~ ~'x, I+A~/
if=0
(2.13)
Now, using Eqs. (2.4b) and (2.9a), the internal forces as defined in Eqs. (2.2) can be written as
N = EAu, x + tINp,
M = EIw, xx + rlMp
(2.14a, b)
Mp  I Apfz dA
(2.1 5a, b)
where Np= ;
A
pfdA,
are the pore pressure resultant and moment resultant over the cross section, respectively. Eqs. (2.14) enable one to express the equilibrium equations, (2.3a) and (2.3b), as
02U ONp 02u EA~x2 + rl~x + qs P~ = 0
04w
o~Mp
o~w
EZ ~x 4  rl Ox2 + qn + P   ~   0
(2.16a)
(2.16b)
The above partial differential equations, (2.14) and (2.16), are not sufficient for determining the six unknown variables, u, w, Np, Mp, N and M, and relations involving fluid flow must be added. Thus the generalized Darcy law which has been accepted as the
Basic Equations
13
benchmark description of fluid motion in completely saturated porous materials (Biot, 1962; Cleary, 1977; Burridge and Keller, 1981; Kingsbury, 1984; Kumpel, 1991; Detourney and Cheng, 1993) is introduced
[k]
=  ~gradpe /zf
(2.17)
where w = (h(u f  uS), and u e and u s are the displacement vectors of fluid and solid material particles, respectively. The dot denotes derivative with respect to time. Here /zf is the fluid viscosity, and [k] is the permeability matrix with components kij. Since fluid flow is taken to be possible in the axial direction only, its only nonzero component is kll, which, by noting that ~"=  d i v w, yields
~ _ kll oq2pf /d,f o~X2
(2.18)
Then by using Eqs. (2.9b) and (2.4b) one obtains
K o2pf
O2w
  ]gf  
Oft
(2.19)
hEzff~x2 + hg~xx
where k,, K   /.Lf13
(_ kllhE)
(2.20)
/ZfT]
In order to express the fluid pressure in terms of the global quantities Np and Mp, Eq. (2.19) is first integrated over the cross section, and is then multiplied by z and integrated over the same area. Noting that w and u are not dependent on the integral variable, these yield
32Np
K
hE
 iVp3x 2

oq2fi; f j
~
a2a//p  K
e2w 
ax 2

l~p

AEI
Oft
A
zdA + A E A ~
3X
aa f
+ hE ~
~
ax
j
A
z dA
(2.21a)
(2.21
b)
where it has been assumed that the order of operations of integration over the area and derivatives with respect to time and axial coordinate can be interchanged. Note that the above two equations are coupled. In order to uncouple the response to axial loads from the response to transverse loads, the yaxis is chosen to be a centroidal axis of the cross section. Thus the integrals above are zero and Eqs. (2.21) are reduced to
K
o72Np
aft
]~[p 3x 2
q A E A ~   0
a2a4p K ~X2
(2.22a)
3x
a2w 
](/lp 
hEI
=
0
(2.22b)
Now the boundary and initial conditions are considered. The boundary conditions on displacement for the axial problem are that u is given at certain points on the beam,
Ch. 2
14
Modeling of Poroelastic Beams
possibly as a function of time. For the bending problem the displacement w and/or the (rotation) angle 0 may be specified, i.e.
Ule= /i,
Wle 1~',
0[e=
0
(2.23a, b, c)
where the subscript e refers to a given point, and an overbar refers to a given function. As mentioned above, for the linear theory used here, 0  W,x. The mechanical boundary conditions for the axial problem are that N is given at certain points, and for the bending problem M and/or Q are specified
Nle/V,
Mle/IS'/,
Qle
0
(2.24a, b, c)
The diffusion boundary condition for a permeable end surface is that Pf[e/3f, while for an impermeable end surface the boundary condition is Pf,x[e 0. In terms of pore pressure resultants, these are
Np[e= ]Vp,
Mp[e/~ p
(2.25a, b)
for a permeable boundary, and
ONe[3xe0,
(2.26a, b)
3MPl3xe  0
for an impermeable boundary. The initial conditions can be determined by considering separately the case of suddenly applied loads, i.e. a jump at t = 0. Then at t = 0, we have ~"= 0 and the relationships (2.12) and (2.13) apply. Eq. (2.12) reveals immediately that the instantaneous response of the beam is that for an elastic beam of Young's modulus (1 + h r/)E. Then u(x, 0) and w(x, 0) are calculated and N(x, 0) and M(x, 0) can be found. (If the problem is statically determinate these would be known a priori.) Then integrating Eq. (2.13) over the area, and also after first multiplying by z, we have h Np  h~ 1+ N,
h Mp = 1 + h~ M,
t  0
(2.27a, b)
These can be given in terms of the displacements as
Ou Np  A E A  
O~X'
OZw Mp   A E I ~
O~X2'
t  0
(2.28a, b)
Thus the pore pressure resultant and moment resultant at t = 0 can be determined. Hence all necessary equations for the problem have been found. In summary, for the axial part of the problem the system of Eqs. (2.16a) and (2.22a), having two unknowns u and Np, must be considered. For the bending part of the problem the system of equations, (2.16b) and (2.22b), having the two unknowns w and Mp, are to be considered. Further, Eq. (2.14a), which additionally contains N, or Eq. (2.14b), which contains M, can be included in the above problems, respectively, if this is required by the type of boundary conditions in the problem considered. Finally, it is noted that if h and r/are zero (if h = 0 then r/ 0, and vice versa), the corresponding equations will degenerate to those for an elastic beam, or a drained poroelastic beam.
Characteristic Times
15
2.2. Characteristic Times Before the equations governing the problem are systematically solved, it is important to consider the nature of the behavior, with respect to time, for certain combinations of the physical and geometrical parameters of the beam. Consider a segment of the beam of length L, onto which is suddenly imposed an axial displacement and thus an initial pore pressure is produced. Now, suppose the segment is restrained in order to maintain the displacement unchanged (i.e. ti  0), while permitting diffusion thereafter through both ends of the beam (at x  0 and x = L). For such a case, it is found from Eq. (2.22a), with the last term on the left taken as zero, that the pore pressure resultant of the segment is given by Np  ~ . b~sinn~rx exp 
n=l
L
LZ/K
t
(2.29)
where the Fourier coefficients bn are determined by the initial pore pressure resultant,
Np(x, 0). Hence L2 ~'D = ~
(2.30)
is a measure of the diffusion time of a beam of length L. This clearly applies also to the situation of bending, i.e. a suddenly applied curvature since from Eq. (2.22b) it is immediately seen that Eq. (2.29) will be valid with Np replaced by Mp. Next, consider a drained beam (i.e. rt = 0) of length L. Taking qn to be zero, Eq. (2.16b) gives the governing equation for the free vibration of the drained beam which is elastic
04w
o~w
E1 ~x4 + p  ~
0
(2.31)
If the beam is simply supported and no moments are applied at the ends, the deflection can be given in the form oo
w = ~" c~sin ~n'nx sin(ant +/3~) L n=l
(2.32)
where a~ = (nTr)2x/EI/pL 4 are then the natural (circular) frequencies of the drained beam. The period of the first bending mode is (2/Tr)~/pL4/EI. Thus the characteristic time for the drained beam is introduced as L4
~'s 
(2.33)
E1
which is the period of the first bending mode times 7r/2. If the fluid is trapped, this must be
pL4 TT
(1 + Art)E1
(2.34)
16
Ch. 2 Modeling of Poroelastic Beams
For later use, a nondimensional parameter is defined as follows:
y _ K _E_i~/P
(_
rs)rD
(2.35)
It should be further pointed out that the actual time required for completing the diffusion produced by an initial pore pressure in a beam not completely restrained is dependent on At/ as well as on 7D. For example, if a simply supported beam with permeable ends is subjected to a sinusoidally varying load qn = sin(rrx/L), which is suddenly applied, the deflection and pore pressure moment resultant prescribed by Eqs. (2.16b) and (2.22b) and the related boundary and initial conditions are found to be W
 77.4~
and
1 
1 q aT]
qnL2(
Mp
exp(
rr2(1 + At/) exp  ( 1 +
(1 qt
I~.'o)L2/Kt
ayl)t2/g t
/]
(2.36)
(2.37)
Thus another characteristic time is recorded rf =
(1 +
Ar/)L2 K
(2.38)
Consequently, three time scales are required to define the poroelastic beam model: rD, the diffusion time when the beam is restrained from further deformation; rf, the diffusion time when the beam is free to deform after a sinusoidal load is suddenly applied; and rf, the characteristic time when the fluid is trapped. On the other hand, rs is a characteristic time for an elastic beam (the drained beam). In practice, however, any three of the four time scales can be taken to describe the system, in which three independent quantities, (1 + At/), LZ/K and pL4/EI, are involved in measuring time.
2.3. Note on a B e a m I m p e r m e a b l e at Both Ends
As is seen from Eqs. (2.27), at t = 0, Np or Mp can be expressed in terms of N or M in a simple way. This of course does not apply to the situation when t > 0. In general, the relationships between Mp and M (Np and N) are prescribed through the differential equations involving the displacements. However, certain relationships can be found in an obvious manner if the beam is impermeable at both ends; these are derived below. According to Eqs. (2.9), the pore fluid increment is given as 
E
a
Pf + rx
(2.39)
Now, when the beam is sealed at both ends and thus no fluid is lost (and since there is no fluid movement perpendicular to the beam axis) it must be that at any time ~ L ~'dx = 0 for any position on the cross section. Hence
Equations in N o n  D i m e n s i o n a l F o r m
f
v
CdV  0
f 1
and
.] v
17
~'z dV  0
(2.40a, b)
Noting that the material parameters, E, A and r/, are at most xdependent, then by substituting Eq. (2.39) into Eqs. (2.40), the following integral relations are found: LE
A
NpN
dx0
(2.41a)
LE
A
Mp  M d x  0
)
(2.41b)
at any t. When t approaches infinity, both Np and Mp will approach constants and thus Np 
~ L[ ~TU/E] dx fL[( 1 + Ar/)r//AE] dx'
t  oo
(2.42a)
Mp 
f L[rlM/E] dx IL[( 1 + Ar/)~'ffAE] d x '
t  co
(2.425)
and when the material parameters are constant along the beam length, the above equation are reduced to
Up
A 1 f N dx, 1 +Ar/L L
A lfMdx Mp 1 + A ~ / L L
t  co
'
(2.43a)
t = co
(2.43b)
2.4. Equations in NonDimensional Form For the sake of convenience for use in the following chapters, in which solutions will be sought, the relevant equations are presented here in their nondimensional forms. The quantities involved are normalized as follows: x*
,_ w
x L ' w L'
t* .
Kt L2,
, Np
.
Np EA'
, qs
qs .L EA ' N*
., qn
N EA'
qn L3
E1 ' , Mp
MpL E1 '
,
u
u
L ' M*
(2.44)
ML  E1
It is required that EA, E1 and K are constant in order that it is meaningful to do this. Then the partial differential equations, (2.14), (2.16) and (2.22), take the forms given below: N*
O~U* Ox*
+ ~?Np
'
092W*
M*=  ~
Ox.2
which represent the constitutive law,
+ r/Mp
(2.45a, b)
18
Ch. 2
On2/,/* 0X .2
o~g;
,
o~X*
Oq4W*
o M;
O~X.4
0X,2
..
072U*
+ qs  ~
+ rl~
Modeling of Poroelastic Beams
. +qn
= 0
Ot*2
+ T2
(2.46a)
2w* Ot*2 = 0
(2.46b)
which refer to the equilibrium of the beam, where the dimensionless constant Yu is defined as Yu
s
(2.47)
EA
y is given in Eq. (2.35), and
02N 
0X.2
Ot*
~02 +A _u~
Ox* Ot*
_0,
0X.2

~ 3a A _w~ =0 Ot* 0X.20t*
(2.48a, b)
The dimensionless forms of the boundary conditions, Eqs. (2.23)(2.26), and the initial conditions, Eqs. (2.27), remain unchanged. However, the initial conditions, Eqs. (2.28), must be rewritten as follows: Ou* Np = A Ox7 ,
OZw * Mp =  A ~
t*  0
09X,2 "
(2.49a, b)
The relations for the sealed beam are converted next. It is supposed that E is constant. Then Eqs. (2.41) takes the following form for 0 < t* < c~:
0
(2.50b)
and Eqs. (2.42) can be written as ,
,
Np =
i01[(1 + Ar/)~//A]dx*'
t  oo
(2.51a)
, Mp =
f~ ~TM*dx* i l [ ( 1 + Ar/)r//A]dx*'
, t  e~
(2.51b)
Finally, by Eqs. (2.43) one obtains
=14
Mp =
An A
I+A~
dx ,
1M* dx*, o
t  oo
t = c~
which requires that A and 77 as well as E are constant.
(2.52a)
(2.52b)
Chapter 3 ANALYTICAL SOLUTIONS FOR QUASISTATIC BEAMS
In this chapter we consider quasistatic solutions for the case of bending only, i.e. qs = 0 and with inertia neglected. All quantities are considered in their nondimensional forms except when otherwise noted. However, for the sake of convenience we omit the superscript * and also the subscript n on the loading function qn. Further, 'pore pressure' is used to refer to Mp when no confusion is thus produced.
3.1. Simply Supported Beams with Permeable Ends Consider a beam subjected to the load q(x,t) only. At t < 0 the beam is unloaded. For the problem considered, the boundary conditions are then (based on the section title specification)
w(O,t) = w(1, t) = 0
(3.1a)
W,xx(O, t) = W, xx(1, t) = 0
(3.1b)
Mp(O,t) = Mp(1, t) = O
(3.1c)
where the second equation follows from Eq. (2.45b) and the third from Eq. (2.27b). The initial conditions are
w(x,O)  wO(x),
Mp(x, O) = M~
(3.2a, b)
If there is no jump in q(x, t) at t = 0 then the righthand sides above are zero; if there is a jump then w~ is taken from the elastic response with Young' s modulus taken as (1 + Ar/)E, and M~p is found from Eq. (2.49b). The assumed solutions are of the form oo
w
E Wn(t)sin(n~rx), n=l
0o
Mp = E mn(t)sin(n~rx)
(3.3a, b)
n=l
which satisfy the boundary conditions (3.1). The functions % ( 0 and mn(t) are to be determined. The load q(x, t) is also expanded in a Fourier sine series in x (assuming this
20
Ch. 3
Analytical Solutions for QuasiStatic Beams
to be possible) q(x, t) = Z bn(t)sin(n~rx),
n=l
bn(t) = 2
(3.4a, b)
q(x,t)sin(nTrx) dx
Then, by substituting Eqs. (3.3) and (3.4a) into (2.46b) and (2.48b), the following pair of equations for determining the tOn(t) and mn(t) are obtained (the inertia is zero): rhn(t) + n 2 ~ m n ( t )  An2~ ~bn(t) = 0
(3.5a)
rln27r2mn(t) + n47r4tOn(t) + b n ( t )   0
(3.5b)
The initial conditions that (3.5) are required to fulfill are obtained by inverting (3.3) and (3.2) for t  0, i.e.
tOn(O)2~iw~ mn(O ) 
(3.6a)
fl
2 J0 M~p(X)sin(n~x)dx
(3.6b)
The system (3.5) together with (3.6) constitutes an initial value problem for each n; the complete solution is then mn(t)=
[
A
mn(O) (l+An)n 2 ~
ft bn(tt)exp( n27r2 ) ) dt tt] o (1 ~ A  r / ) e x p
('
n27r2 1 ~ At/t (3.7a)
tOn(t) 
bn(t) rl n4 7r4  n277.2mn(t )
(3.7b)
3.2. Beams Subjected to Loads Suddenly Applied and Constant Thereafter
We will limit ourselves to cases when M is not time dependent for t > 0, as is obviously the case for statically determinate beams. We have already pointed out that at t  0 the problem is solved by finding the solution to the elastic problem with Young's modulus taken as (1 + Art)E. Consider now the long time solution (t ~ co). Fluid motion in the axial direction ceases and derivatives with respect to time approach zero. pf, and by its definition also Mp, necessarily become independent of position along the length of the beam. If at least one of the beamends is permeable Mp approaches zero and the equations simply reduce to the elastic beam equations with Young's modulus being that of the drained beam, E. So the long time deflection is easily found. If neither end is permeable then the pore pressure will become constant, and can be determined by Eq. (2.52b), oo
MP = 1  / A n
0
M(x, co) dx
Beams Subjected to Loads Suddenly Applied and Constant Thereafter
21
Even if both beamends are impermeable, for the special condition that the integral above is zero, i.e. ~~0M(x, oo) dx = 0
(3.8)
there is no long time pressure in the pores and the elastic solution with Young's modulus taken under the drained conditions is the solution for long times. For all cases when Mp 0 we can generally find series solutions for the transition behavior between the initial response and the long time response. Let the solution for deflection be written as
w(x, t) = wE(x) + Aw(x, t)
(3.9)
where w E is the elastic solution of the drained beam. Then M   w E , XX " Substituting this relation into (2.45b) and rewriting (2.48b) yields
02Aw r/Mp 
o3x2 ,
~Mp _ ~tp + A 02~ * o3x2
( 3 . 1 0 a , b)
o~x~
By separating variables, a general solution is obtained in the form
( 1
Aw = [cl + c2x + c3cos(cox) + c4sin(cox)]exp 
Arlt
)
(3.11)
Thus, if the constants ci can be determined so that the boundary and initial conditions are satisfied, the solution is found. In the following, Aw will be determined for various cases. Aw will take the form of sums of solutions of the type given in (3.11) in order to satisfy the initial conditions of a particular problem. This is justified since the partial differential equation for Aw is homogenous. Since w E necessarily satisfies the boundary conditions required for w, the boundary conditions for Aw will always be homogeneous. The initial condition on Aw is simply the difference between w(x, 0) and w E, i.e. AwIt=0~  A'~~wE
(3.12)
I+A~
A simply supported beam with both ends permeable The boundary conditions for this case are Awls0 Awl~=l Aw, x~lx=O A W ,
xxlx=l = 0
The last two conditions follow from Eq. (3.10a) by noting that Mp(0, t) = Mp(1, t) = 0. The various ci are obtained as cl = c2 = c3 = 0, so that
Co = nor
(n = 1,2, 3 .... ),
c4 nonzero
Ch. 3 Analytical Solutions for QuasiStatic Beams
22
Aw = Z
bnsin(nTrx)exp 
1 + hr I
n=l
t
(3.13)
is an acceptable form for the solution. Comparing this form at t = 0 to the initial condition on Aw we have co
bnsin(nerx)
Z n1
I~~wE
(3.14)
1 + At/
and the constants bn can be found by expanding wE into a Fourier sine series.
A simply supported beam with
one end
impermeable and
the other end
permeable
The boundary conditions here are
Awlx=oAWlx=l
03â€˘ ~
_ ~Xw[ x=0
0
~X"~x=l,
The third condition follows from Eq. (3.10a), when noting that Mp,~(0, t) = 0. Aw is found to be of the form
[
(2m 1
~ . bm  1 + x + c o s ~ 7 r x
Aw
m1
exp 
2
4(1+A~/)
t
(3.15)
Thus, if the series below is uniformly convergent, we can write
OAw (x, O)  mZ1b m 1  2 m 
Ox
=
2
1 zrsin
2m1 2
\I 7rxl[ )'_1
(3 16)
and therefore oo
Z
bm
OAw Ox (0, O)

(3.17)
m=l
For the determination of b m a n odd function ~O(x) with period 4 is introduced, which is defined in the interval [0,2] by ~x)

f   ~OAw x ( x , 0 ) + OAw (0,0), 3x
~2x),
if0 0 There are two conjugate complex roots in this case
sc2,3  ao /3oi where ao and 13o are real and given by
(5.29a)
Ch. 5
58 ce~
A +2 B + 13 ] (nTr)2'

Vibrations of Poroelastic Beams (5.29b, c)
Vc~A 2 B (n,n.) 2
/3~
The only real root ~1 is given by Eq. (5.2 l a). The denominator of Eq. (5.10) can then be rewritten as
( s  ~ ) ( s  ~2)(s ~3) ( s  ~)[(s c~o)~ + ~3~]
(5.30)
and Eq. (5.10) becomes
kl ~1
COn(S)  
~
11

kz(sCeo) ( S  ~ o ) 2+~3~
k3
(s
)2
(s~o
12(S CeO)
+
 ~1
"~
+t~o~
t3 )t;.(s) + ( s  ~o)2 +/302
o~o)2 + ~
(5.31)
where the constants are given by k1
k2 
k3 z
/3 + ~
+ O~n(0)~
( ~  ~o) 2 + t~ /3
Ce~l + (ce~ + /32  20~0~l)tOn(0 )
(r  ~o)2 +/3~ ('~o  ~,)t~ + (~o~ + t~o~  ~or (~
+ [(t~o~  ~o~)r + (~o~ + t~o~)~o]o~(o) .o)2 +/32o
ll
~1 + (n'rr) 2 ~t2[(~ 1 _ OLo)2 _q_ ~2] 
13 =
c~2o + 13o 2  Ceo~:1 + (olo scl)(nzr) 2 ~[(~:1  ao) 2 + 13o 2]
12
(5.32)
Taking the inverse transform and applying (C.2), the solution for tOn(t) is found
tOn(t) kl e~It q kee~~  le[c~176
k3
t) + ge~~ Po
t)  lie ~t
~t
bn(~')e ~IT d~"
o
l23[sin(/3ot)I n  cos(~ot)In]e ~~ ~o
+ sin(~~176
(5.33)
where Ic~ and I n are integrals, defined as follows:
Inc(t) 
0
bn(~')e~~
d~',
I2(t) 
0
b~(~)e~~162
d~
(5.34)
With %(t) in hand, ran(t) is then found from Eq. (5.6a). Thus solutions of the form (5.3) have been found for all cases. Note that the sign of A, and hence the nature of the behavior of the beam is dependent only on A 77 and y.
59
Initial Value Problems
Fig. 5.1. The TA r/plane showing the region where no oscillatory motion is possible (shaded). The coordinates of the cusp of the nooscillation area are shown.
In Fig. 5.1 the shaded area is the region where A < 0. Thus no oscillatory motion is possible for combinations of A r / a n d 3/there. In the language of damped elastic vibrations this is the over damped region. The boundary of the shaded region represents the case of A = 0. In the language of damped elastic vibrations this is the situation of critical damping. Outside the shaded region A > 0 and oscillatory motion is possible. In the language of damped elastic vibrations this is the region of light damping. Now recalling the definitions of the characteristic dissipation time, ~'D, the characteristic time of vibration of the drained beam, ~'s, and the characteristic time of vibration of the beam with fluid trapped, ~'T, we see that ~s _ ~/1 + An, ~'T
~s _ T, TD
~'T _ 3' ~'D ~/1 + An
(5.35a, b, c)
Thus the nature of the solution could have been presented as depending on any two ratios of characteristic times, ~'S/~'T, ~'S/~'D and ~'T/~'D. In practical systems it is unlikely to be situated in the shaded region of Fig. 5.1, and therefore our interest when considering examples will center on the outer region.
Free vibrations of a simply supported beam with a sinusoidal initial shape As the first example, consider a simply supported beam with the following initial conditions: w ( x , O ) = sin(vrx), w ( x , O ) = O , and Mp (x, O)  O. Namely, the beam is displaced into a sinusoidal shape, held still until pore pressure is reduced to zero, and
Ch. 5
60
Vibrations of Poroelastic Beams
then released. We will also consider the situation when after displacement, the beam is released before the pore pressure has had time to decay at all, i.e. Mp(x, 0) is found from Eq. (2.49b). These two cases of initial pore pressure are denoted as I and II, respectively. The solution w(x, t) for 'light damping', the outer region in Fig. 5.1, is given by Eqs. (5.3a) and (5.33). For the initial conditions being considered, only one term of the series is required, and the solution becomes
w(x, t) 
kle ~'t + k2e'~~
t) + ~o e~~
t) sin(Trx)
(5.36)
where ki, ~1, Ceo and/30 are given by Eqs. (5.32), (5.21a), (5.29b) and (5.29c), respectively. The exponents ~1 and a 0 are negative;/30 is positive. ~:1, a0 and/30 depend only on A r/and y. There are thus two exponential decay processes occurring in the system; one results in a monotonic decay of part of the initial deflection, the other multiplies the time dependent sinusoidal oscillations. The relative importance and interaction of the two decay processes is dependent on the values of the k i, which in turn depend in a complicated fashion not only on the system parameters, but also on the initial conditions. Quite interesting patterns are possible. Consider first the simplest situation. If At/is very small (i.e. TS/~'Tis close to 1) or if y is very large (i.e. ~'S/TD is large) it is found that the first term in the square brackets is small compared to the terms involving sine and cosine. This is to be expected since At/ small implies that the presence of trapped fluid does not influence greatly the stiffness of the 1.0
I ,,
Condition I :Solid line Condition II: Dashed line
[ [
I
0.5
0.0
'
,
0.5 Il
I.0
0.0
2.0
4.0
6.0
8.0
t Fig. 5.2. Vibration pattern, w(x,t) at x = 0.5; Initial conditions w(x, 0)  sin(Trx), w(x,0) 0; AT/ 0.25 and y = 1.5.
Initial Value Problems
61
beam; there is little coupling between (2.46b) and (2.48b). Similarly, large Y implies that the period of the drained beam is very large compared to characteristic pressure dissipation time, so with a very small portion of a single period, pore pressure is lost and the beam vibrates essentially as a drained structure. But there will remain some damping, which is not at all insignificant. Typical deflections vs. time patterns in this circumstance are shown in Fig. 5.2 for conditions I and II of initial pore pressure; there is very little difference between the two deflectiontime curves. Note that time has been nondimensionalized with respect to ~'D,the dissipation time. The behavior is similar to that obtained for damped oscillations of an elastic beam. For these conditions the frequency of the system is identified as fi0/27r; using Eqs. (5.16) and (5.29), it is given explicitly as
f
~
4
~+
x/~+~
~
Consider next cases for which the contribution of the first term in the square brackets is not negligible. While for the previous case the frequency of vibration was simply flo/27r, when the first term in the square brackets is not negligible no frequency can be defined. But it is convenient to define a pseudo frequency by
fpseudo /3o127r
(5.38)
This is the frequency at which the system would vibrate if the first term in the square 5.0
4.0
~.~
3.0
0.40'
+ 2.0....
0.60
i 1.0
" . . . .
0.0
i
1.0
. . . .
,
. . . .
2.0
y 141+ x,7
I
.
.
.
.
3.0
(:
Fig. 5.3. Normalizedpseudo frequency map in the ~'T/~'D~'S/~'Tplane.
4.0
62
Ch. 5
Vibrations of Poroelastic Beams
1.0 k..
0.5
0.0 =
o Condition I o Condition II
0.5
1.0
,
,
0.5
0.0


,
t
1
i
1.0

,

1.5
Fig. 5.4. Vibration pattern, conditions as for Fig. 5.2; Condition I refers to initial pore pressure equal zero, and Condition II refers to initial pore pressure as developed immediately after rapid displacement of the beam. At1 15 and y = 1 (ZS/ZT  4 and ZT/ZD = 0.25).
brackets were negligible. In Fig. 5.3 a map of constant values (solid lines) of the combination O is shown, where 0
2 y ~r ~/1 + a v fp~ua~
7r To
seudo
(5.39)
is plotted in the plane of y/~/i + Art vs. x/1 + Art (i.e. ~'T/ZD vs. ZS/ZT). Where the curves are nearly horizontal the behavior is as in Fig. 5.2, and the pseudo frequency is very nearly the frequency of the drained beam (O is asymptotic to 1/,,/1 + Art). The time scale of the beam vibration is sufficiently long with respect to that of the dissipation process so that the coupling through Art is of secondary influence in determining frequency. On the other hand, when the curves are nearly vertical the pseudo frequency is very nearly that of the beam with trapped fluid (O is asymptotic to ~/1 + Art/y). But now the initial conditions very strongly influence the relative importance of the two decay processes. Consider Fig. 5.4 where Art = 15 and y = 1 ('/'S/TT = 4 and 7"T/T D = 0.25). For the initial condition I, the monotonic decay process dominates and the sinusoidal process just adds a small tipple which rapidly disappears. For the initial condition II, it is clear that it is the sinusoidal decay which dominates. The practical implication is that for the first initial condition (I) at the time t  1 initial deflection has been reduced by only onehalf, whereas for the second initial condition (II) the deflection has essentially been
63
F o r c e d H a r m o n i c Vibrations
reduced to zero. This illustrates the crucial influence of initial conditions on the subsequent decay of vibrations. Finally, we point out that when Q  0 the two exponents are equal, and ~:1  a 0  ~'2/3, which yield identical decay rate for the two processes. This occurs when 2y 2  9A~/+ 18  0
(5.40)
and is shown in Fig. 5.3 by the dashed curve. To the left of this curve the sinusoidal decay process is faster than the monotonic process (see Fig. 5.4); to the right of the curve the monotonic decay process is the faster one.
5.2. Forced Harmonic Vibrations
Consider now a beam subjected to the harmonic load (5.41)
q(x, t) = ft(x)exp(iwt)
where i  x/"Z1. Then the forced part of the solution has the form w(x, t)  ~ ( x ) e x p ( i w t ) ,
(5.42)
Mp(x, t)  Mp(x)exp(ioot)
Substituting these forms into (2.46b) and (2.48b) it is found that the amplitudes ~(x) and Mp(x) are governed by the following system of ordinary differential equations: d4~ d2/~p dx 4  T] dx 2 1 q  3/2(.021,~  0,
dZ/l~/p dx 2

iw/f/p

d2ff: iA~o~f  0
(5.43)
Eliminating/f/p yields d6w
dx 6  i w ( 1
d4w 2 d2~ d2q  imP/ 0 + A~/)~~  y z c o ~ + i~/2033w +
(5.44)
For a simply supported beam with both ends permeable the boundary conditions are given by Eq. (5.1). For this case the assumed solution is of the form co
 ~ Wnsin(nTrx) n=l
(5.45)
By expanding the load amplitude into a Fourier sine series oo
g /  ~ . cnsin(nTrx)
(5.46)
n=l
where Cn   2 ~ i ~(x)sin(n~rx) dx
so that the coefficients Wn can be determined from Eq. (5.44) as follows:
(5.47)
64
Ch. 5
Vibrations
of Poroelastic
i~o] Ar/)(n'n) 4 + T2o92(nTr)2 + i'y2o93
Beams
Cn[(nTr) 2 + Wn  

(nTr) 6 
ioJ(1 +
(5.48)
and Mp(x) can be found from Eq. (5.43) in the form (3O
(5.49)
iVlp   ~ . m n s i n ( n ' n ' x ) n=l
where Aoo(nTr)2 mn =
0 9  i(nTr) 2
(5.50)
wn
Such a series solution would have been more difficult to obtain had we considered more complex boundary conditions, e.g. boundary conditions that are not homogeneous. In such situations, a general solution to Eq. (5.44) would be required. The general solutions of the homogeneous differential equation corresponding to Eq. (5.44) would have the form exp(r Here Ck is the root of the characteristic equation 6
 io~(~ + A n ) ~ 4 
~0~2~
+ i~o
3 
0
(5.51)
This equation is changed into the cubic equation (5.13) by letting
10o
10' Q)
It.,
I
0_2
~
~ xn=o.o o xn=o.,
\\
lff" lff'
0.0
6.0
12.0
lB.0
24.0
Y Fig. 5.5. Deflection amplitude vs. frequency of harmonic load; first two resonance areas shown; y = 0.75" At/= 0 (elastic), 0.1 and 0.4.
Closure
65
2
i0)
~
sZ.
(5.52)
Thus roots r (k = 1,2 ..... 6) can be found for all cases. When A # 0, all roots are distinct, and the general solution of Eq. (5.44) has the form
6 ~ ( x )  ~. Bkexp(~kx) + l'~par(X) k=l
(5.53)
where B~ are constants which can be determined according to the boundary conditions, and Wpar(X) is a particular solution of the nonhomogeneous equation (5.44). The following function can be taken as a particular solution if ?:/(x) is a polynomial of third order or less
WPar(x) 
,~(x)
3/20)2
(5.54)
For a more general loading, Eq. (5.45) can serve as a particular solution.
A simply supported beam subjected to a uniformly distributed harmonic load Consider a simply supported beam subjected to a uniformly distributed harmonic load (magnitude is 1, dimensionless), where the forcing frequency i s f = 0)/27r. The rs/rD (=3') ratio is 0.75. Fig. 5.5 displays the first two resonance areas, the first of which can be predicted by Eq. (5.37). The curve for A t / = 0 corresponds to the (drained) elastic case. For f   0 the solution is that of the quasistatic problem. Note that both the first and second resonant frequencies increase as the influence of the presence of fluid is increased ( A t / > 0); the shift is more prominent at the second resonant frequency. On the other hand it is clear that the amplitude response is significantly reduced as At/increases. We see that the amplitude is smaller than the static elastic value if the frequency of the loading is far from the resonance areas. In fact at very high frequencies essentially no fluid movement can take place at all and the beam behaves as an elastic beam whose Young's modulus is
(1 + An)E. In Fig. 5.6, T has been increased to 3; the behavior pattern is similar except that now the frequency shift is nearly imperceptible.
5.3. Closure A detailed solution has been carried out for the simply supported beam with permeable end surfaces. For free vibrations the nature of the deflection decay is strongly affected by the initial pore pressure. Whereas for the damped elastic beam (subcritical) there is always some oscillatory motion, in our case this oscillatory motion is often completely wiped out by the existence of a monotonic decay term. It is seen that axial fluid diffusion in vibrating beams can easily provide a very strong damping mechanism. The mechanism might possibly be influential in reducing damage to plant elements when these are subjected to suddenly applied and short term loading. It is also possible that such a mechanism could be introduced into 'smart' elements in structures; by 'valving' the
Ch. 5
66
Vibrations of Poroelastic Beams
16'
102
tR
o X~=0.1 o X~=0.4
' ~
a),~=l.0 .
10 3
10 4 0.0
1.5
3.0
4.5
6.0
/ Fig. 5.6. Deflectionamplitude vs. frequencyof harmonic load; first two resonanceareas shown; y = 3" A~7 0.1, 0.4, 1.0 and 3.0.
diffusion boundary condition at the element end(s) one could rapidly change the response characteristics of the system, and thus provide an efficient control arrangement. Finally we point out that ), is inversely proportional to the viscosity of the contained fluid (see Eqs. (2.35) and (2.20)). Viscosity of most fluids is very sensitive to temperature; for instance a change in water temperature from 10~ to 40~ reduces viscosity by one half. Then considering Fig. 5.3 it is immediately apparent that such a temperature change in water saturated material can completely change the regime of vibration response of a structure. It is noted that structural damping is often related to and modeled as resulting from the viscoelasticity of the material in which case two time scales are involved, one which is related purely to the material, the second which also includes structural parameters. In the present model three time scales are involved. Additionally, in the present model a new type of initial condition must be included which can appreciably influence behavior patterns. For all of these reasons behavior patterns are possible which are not seen in viscoelastic structures.
Chapter 6 LARGE DEFLECTION ANALYSIS OF POROELASTIC BEAMS
In the previous chapters, the deflection was considered to be small and thus linear theory is sufficient since the constitutive law adopted is linear. Yet for some situations, it may be necessary to employ a large deflection theory in order to correctly describe the behavior of the porous components. The deformation can still be small and elastic; the large deflection is possible because of the slenderness of the components. Therefore, the components are modeled in this chapter as geometrically nonlinear (but constitutively linear). Biot's constitutive law and Darcy's law are then still applicable, while new geometrical relations and equilibrium equations must be introduced.
6.1. Governing Equations As in all previous analyses, the beam considered is taken to be transversely isotropic in the crosssectional plane and the microgeometry of the bulk material is such that the fluid flow is possible in the axial direction only. The basic assumptions for the small deflection theory are still justified for our problem, i.e. crosssections remain plane after deformation and the shear strain and the transverse normal stress are negligible. Two more assumptions, which are implied in the small deflection problems, should be restated here: (1) the beam has a principal plane and bending occurs within the plane (i.e. the xz plane, shown in Fig. 6.1), and (2) the deformation is small so that Biot's constitutive law is justified. A segment of the beam axis with an original length &x and deformed length As is shown in Fig. 6.1. s is always measured along the beam axis during the deforming process; 0 is the rotation angle of the crosssection, which may be large; u is the horizontal displacement and w is the vertical deflection of the beam axis. The area and moment of inertia (with respect to y) of the crosssection are A a n d / , respectively. N, M and Q are the resultants of the normal stress and shear stress at the crosssections produced by the distributed axial load qs(x) and normal load qn(x). The directions of the two loads are taken to be unchanged after deformation, but their magnitudes are generally functions of x and t. (It is important to remember that x in this chapter always refers to a material point along the beam.) The geometry of the beam is considered first. Using e0 to represent the strain of the axis, we have
68
Ch. 6
Large Deflection Analysis of Poroelastic Beams
q.
a l..
.I b
N M
'
I 
 ~
N+AN
Fig. 6.1. Free body diagram of beam segment.
As(1
+ e0)Ax
(6.1)
Since the higher order of infinitesimal is omitted, Au + A x  As cos0 and Aw  As sin0 hold according to the geometrical relationships. Thus, using Eq. (6.1) the following transformations between the two sets of variables: u, w and e0, 0 are obtained: 3u 3x
(1 + e 0 ) c o s 0  1,
Ow 3x
  ( 1 + e0)sin0
(6.2)
The geometrical equation, i.e. the relation between the strain and displacement is e:
30 e0Z3x
where ~ is the normal strain of the solid matrix in the direction of the beam axis. Since the strains are small, the following relation is used instead of the above expression in order to simplify the formulations: 30 e = e0  z  3s
(6.3)
The difference thus produced is z(30/3S)eo which is a higher order term as compared with both e0 and z(30/3s). The constitutive relations for the beam are as introduced in Chapter 2. 7 = Ee
~pf,
~"= r/e +/3pf
(6.4a, b)
The normal stress ~ at the crosssection leads to the integrated resultants N and M (see
Governing Equations
69
Eqs. (2.2)) as shown in Fig. 6.1, which can be given as follows by Eqs. (6.4a) and (6.3)
N
EAeo + rlNp,
M
 E l O0 + riMp 0s
(6.5a, b)
where Np and Mp are again defined as
fpfdA,
P
Mp=~pfzdA
A
(6.6)
A
and the yaxis is taken to be the centroidal axis of the crosssection so that IA z d A = O. The equilibrium of the deformed segment as shown in Fig. 6.1 requires E
Mb' = 0:
(M + zkM)  M  QAs + o(As) = 0
~. Fz = 0:Nsin0 + Qcos0 +
qnZ]kx

(N + z~)sin(0 + A0)  (Q + AQ)cos(0 + A0) = 0
E Fx = 0:  Ncos0 + Qsin0 + qsZLr+ (N + zSaV)cos(0+ A0)  (Q + AQ)sin(0 + A0)  0 in which o(As) represents the second or higher order of the infinitesimal As. The first equation shown above is simplified as follows when o(As) is omitted:
OM Os
 O
(6.7)
Further, multiplying the second equation by cos0 and the third by sin0 and then adding together, yields Q + (qncos0 + qssin0)zkr  (N + Z~)sin(A0)  (Q + AQ)cos(A0)  0 If the curvature O0/Os is not infinite, namely a moderately large deflection situation is considered, then A 0, 0 when As , 0. For this case then, sin(A 0)  A 0 and cos(A 0)  1. Moreover, ANAO is negligible since AN is of the same order as A0. Thus, the above expression leads to the following differential equation for equilibrium:
OQ oO  qnCOS0 + q s s i n 0  N  Ox Ox
(6.8)
Similarly, another differential equation for equilibrium can be obtained
ON O0  % s i n 0  qscos0 + Q ~ 0x 0x
(6.9)
Eqs. (6.7), (6.8) and (6.9) are all the differential equations required for equilibrium of the beam. Since Q has no relationship with the diffusion, it can be eliminated from (6.8) and (6.9) by using Eq. (6.7). Noting also Eq. (6.1), the two differential equations required for the problem in hand are qnCOS0+ qssinO
02M 
ON 3s

l+eo
o~S2 qnsin0
qscosO
OM O0 +
l+eo
O0 Os
N
3s Os
(6.10)
Ch. 6
70
Large Deflection Analysis of Poroelastic Beams
which, after (6.5) is introduced, can be written in new forms
El
03 0 02Mp ~0 qncosO + qssinO ~ + 7q Os2 + (EAeo + r/Np) Os 1 + e0 (6.11)
020 E z gj
OMp ) O0  n   ~s
~
3Np + EA Oeo _ qnsinO qscos0 + n
~s
~s

1 + ~o
Now the fluid flow physics must be considered in order to complete the system of differential equations for the problem. This is governed by Darcy's law, which (cf. Eq. (2.18)) takes the following particular form when only the axial diffusion is considered: ~=
kll O2pf
(6.12)
]d~f o~S2
where kll is the permeability in the axial direction and ~f is the viscosity of the pore fluid. Substituting Eq. (6.4b) and then Eq. (6.3) into the above, the form below is obtained: 02pf
00
(6 13)
K ~s~ = pf + A E ~ 0  AEz~s
and again K  kll AE/(/zf r/). If differential equations expressed by the pore pressure resultant, Np, and pore pressure moment resultant, Mp, instead of the local pore pressure, are sought, the operators
fA [ ]dA'
fA [ ]zdA
should be applied to Eq. (6.13) respectively, yielding
K
o?2Np  Np  AEA g:o, 3S2
K
o?2Mp 00  Mp + A E I ~ o~S2 o~S
(6.14a, b)
Eqs. (6.5), (6.10) and (6.14) are all the governing equations necessary for the six unknowns" e0, 0, Np, Mp, N and M. Alternatively, only Eqs. (6.11) and (6.14) will be involved if only four unknowns" e0, 0, Np and Mp are considered  when the boundary conditions permit us to do so. Note that as opposed to the situation for the small deflection problem, here the axial variables and the bending variables are inextricably coupled. Finally, the boundary conditions and the initial conditions for the problem are considered. The geometrical boundary conditions are Ule= / L
Wle ~,
0le = 0
(6.15)
where le represents that a variable takes on a specific value at a boundary, and a bar refers to a given function. No boundary conditions are required for e0 since none of its derivatives with respect to x or s are included in the differential equations. The mechanical boundary conditions are the same as those for the poroelastic beam with small deflections. The load conditions at boundaries are Nle iV,
Mle=/9/,
Qle = 0
(6.16)
E q u a t i o n s in N o n  D i m e n s i o n a l
71
Form
The diffusion for permeable boundary conditions are
Up]e ~p,
Mp]e
Mp
(6.17)
and for impermeable boundary conditions they are
ONe Io~Se0,
0MP loS e   0
(6.18)
The initial conditions for e o, 0, Np and Mp are dependent. The relationship is determined by the requirement that ~']t=o0, which means that there is no instantaneous diffusion. Using Eqs. (6.4b) and (6.3), this becomes 00 pf
AEz AEeo, Os
(6.19)
t0
and upon substituting into (6.6), the equivalent expressions in terms of the global quantities are derived 0O 0s'
Np  A E A e o ,
Mp  AEI~
(6.20a, b)
tO
6.2. Equations in NonDimensional Form when t~0  0 A particular but very likely situation is considered in the following. This is, when the axial strain is negligibly small, as is to be expected for a slender beam. Letting e0  0, then, from Eq. (6.1), As = Ax. The differential equations in dimensionless form are obtained after the quantities are normalized as
x
, = _x L'
t* _
,
MpL
Mp
E1
'
Kt L2,
, _ u
,
qn L3
qn
E1
u L'
'
, _ w ,
qsL 3
qs
E1
w L'
N* _
NL2 El'
M* = ML El'
(6.21) where L is the length of the beam. In accordance with Eqs. (6.2), (6.5b), (6.10) and (6.14b), the relevant equations are as follows" 0u* Ox*
= cos0  1,
0w* Ox*
= sin0
(6.22a, b)
which are the geometrical relations; M*
00 0x*
+ r/Mp
is the constitutive law in terms of the global quantities;
(6.23)
72
Ch. 6 ,
, .
O2M*  qncos0 + qsslnO  N* 00 Ox.2
Ox*'
Large Deflection Analysis o f Poroelastic Beams ,
ON* _ q~sin0 qsCOS0 Ox*
0114" 00 Or,* Ox*
(6.24) are the equilibrium equations; and 32Mp
00
0X.2 /~'/; nt/~0X*
(6.25)
derives from Darcy's law. A dot now refers to a derivative with respect to the dimensionless time t*.
The dimensionless forms for Eqs. (6.5a) and (6.14a) are not included here since they are not relevant after e0 is omitted. N* must be determined instead by the equilibrium equations, as given by Eqs. (6.24). The forms of the boundary conditions (6.15)(6.18) remain unchanged after the normalization. The initial condition, according to Eqs. (6.20b) and (6.5b), can be written as 00 M*  ( 1 + At/) 0x*'
,
t 0
(6.26)
or in another form ,
A
M p   A1 +~
M*
'
t*  0
(6.27)
Either of these will be helpful in computing initial values. The system of partial differential equations above is complete. It should be noted that certain unknowns may have to be solved simultaneously. For example, although the deflection w* appears only in (6.22b) and it can possibly be calculated separately after 0 is obtained, in general cases w* must be solved simultaneously due to coupling of the boundary conditions. For instance, there are two boundary conditions for w* but no boundary conditions for 0 for a simply supported beam. Elimination of N* and M* from the simultaneous equations is also possible but it will increase the order of the differential equations, which usually results in precision loss in numerical computation and also results in inconvenience for introducing boundary conditions. Therefore, the five equations (6.22b) and (6.23)(6.25) will be solved simultaneously for five dependent variables, N*, M*, Mp, 0 and w*. Nevertheless, when computing the initial values for the variables, only Eqs. (6.22b), (6.24) and (6.26) need to be solved simultaneously for N*, M*, 0 and w. Then Mp* can be calculated by Eq. (6.27). On the other hand, u* can always be calculated separately by Eq. (6.22a) after 0 has been solved since its boundary condition is not coupled with others in our situation.
6.3. N u m e r i c a l F o r m u l a t i o n
Analytical solutions for the system of nonlinear partial differential equations are not expected to be obtainable. Numerical solutions for these equations are derived in the following by employing the finite difference method for the spatial coordinate. The equations are employed in their nondimensional forms; the superscript * will be omitted for convenience.
N u m e r i c a l Formulation
73
The numerical algorithm is presented now. The beam is discretized by nodes x i (i = 1,2, 3 ..... m). These nodes are not necessarily uniformly distributed and the domain is not necessarily from x = 0 to x = 1, but it is convenient to start from x  0. Accordingly, the subscript i is used to denote that a variable is valued at x = xi. Discrete times are denoted by tj (j = 0, 1,2, 3 ..... n and to = 0); also the time intervals are not necessarily equal. A superscript j then implies that a variable is valued at t   t j . The difference formulas derived by the Lagrange interpolation are adopted. The 3node formulas for the first order derivative at the node i, for example for M, are given as
( 3M)Ji CoM~ i j + C1M~+ i j 1 ._[_ C2Mi+ i j 2
(6.28a)
( 3114 )i_
(6.28b)
/(3114 ,,1)i ~x

ij
C ilM~
i
+ CDIldj + C1M~i+ 1
j
C  2M~_ 2 +
Ci
J at C~M~ 1M~I
(6.28c)
where the coefficients C i, which depend on the coordinates of the node i and its two neighboring nodes, are given in Table D. 1 of Appendix D. Eqs. (6.28a,b,c) are the forward, central and backward difference formulas, respectively. The forward and backward formulas are used only at the end nodes; otherwise the central formula is adopted. In this way, the matrix obtained for the algebraic equations is banded and the bandwidth is five (pentadiagonal; tridiagonal if the first and the last equations are excluded). Alternatively, for any node i which is at least two nodes away from the end nodes (i.e. when 3 ~< i ~< m  2), a 5node central difference formula can be applied 9
2
=
C~M~+~
(6.29)
k=2
where the coefficients C i are given in Table D.2 of Appendix D. This does not change the bandwidth but may improve the accuracy of the computation. In practice, one needs first to calculate the initial values for all variables by the initial condition (6.26). Eqs. (6.22b) and (6.24) are necessarily included in order to determine N, M, 0 and w at t  0. Necessary reforms to these equations are to be made for the numerical computation. Replacing O0/Ox in (6.24) by M, using (6.26) and letting Q  OM/Ox, the ordinary differential equations for time t  0 are dw dx dO dx dM dx
= sin0
(6.30a) M

l+h~/  Q
(6.30b)
(6.30c)
Ch. 6
74
Large Deflection Analysis of Poroelastic Beams
dQ MN  qncosO + qssinO + dx l+Art
(6.30d)
dN MQ  qnsinO qscosOclx l+Art
(6.30e)
Together with the corresponding boundary conditions, the initial values can be determined. The shear force Q is involved in the numerical computation in order that all equations in (6.30) have identical forms and thus the same procedure in computation can be followed. This also helps with manipulating a boundary condition with a given shear force. The system of equations for the finite difference method can be obtained after (6.28), or alternatively (6.28) together with (6.29), are applied to each equation in (6.30). For example, the following apply for (6.30c) if only the 3node formulas are used: i
i
Ciomi + c1mi+l + c2mi+2  Qi, i
C i l M i  1 1 CoMi + C1Mi+l  Qi, i
c i 2 m i _ 2 k C i l M i _ l + C'om i  Qi,
for i   1 for 2 < i < m 
1
(6.3 l c)
for i   m
where all variables are valued at to and thus the superscript j ( = 0) is omitted. We will continue to do this when no confusion will be caused. Eqs. (6.3 l a,b,d,e) are omitted here since they have similar forms to (6.31c); only the righthand sides are different. Thus 5m nonlinear algebraic equations are involved, which are linearized, assuming that the righthand sides are known. Since iteration is unavoidable in solving for the nodal values, each set of the algebraic equations (e.g. (6.31 c) alone) might as well be considered as an independent linear system. Every system of equations has the same coefficient matrix (m X m with bandwidth 5) before the boundary conditions are involved. The solution procedure is as follows: 1. Give initial values to Oi, Mi and Ni (an initial guess) in order to start an iteration; 2. solve the linear equations (6.31d) for Qi, (6.3 lc) for Mi, (6.31b) for Oi and (6.31a) for wi successively (always using the newest results to compute the right hand sides of the equations); 3. finally solve (6.31e) for Ni; 4. repeat steps 2 and 3 until the desired accuracy has been reached. An advantage of this scheme worth noting is that only in solving for Qi do we require the initial guess or the results from last round of iteration. This certainly helps the convergence of the algorithm. In an iterative procedure, the initial values for starting an iteration should be carefully chosen in order to guarantee a convergent result. In our situation, when the loads are not too high, the 'small deflection solution' (formally obtained from Eqs. (6.30) by letting 0Ow/Ox, s i n 0   0 , c o s 0 = 1 and deleting the coupling items MN/(1 + Art) and MQ/(1 + Art)) should be a good choice. However, this results in divergence when the loads are sufficiently high since the 'small deflection solution' deviates too much from the
75
Numerical Formulation
sought after solution. Nevertheless, the loading can alternately be performed step by step in the computation. In the first loading step, very small loading values are taken so that the small deflection solution can be justifiably adopted. Alternatively we can adopt a zero solution when the first step loading is taken to be zero, although the cost is higher. In each following loading step, the scheme stated above is applied with the loading values being increased by a proper increment and the results from the last step being used as the initial guess. The results are expected to converge to the solution for the required load after the loading values are finally increased to these values. Note that for a cantilever beam, there exists one boundary condition for each of the five unknown variables (w, 0, M, Q, N). Hence no difficulty occurs when they are solved separately. Otherwise, for instance, if there is no boundary condition for Q (when there are two for w, as encountered in a simply supported beam), the values of Qi cannot be determined in each step. In other words, w, 0, M and Q should be to some extent solved dependently for general cases (N has an independent boundary condition in all cases considered). Therefore, step 2 should be modified. One can always first suppose wl, 01, M1 and Q1 to be zero and solve for Qi, Mi, O i and W i successively. Wl, 01, M1 and Q1 could be determined by four simultaneous algebraic equations given by four boundary conditions and then Qi, mi, Oi and wi can be modified to satisfy the boundary conditions. This can be done in a straightforward way for a linear system (see Appendix E), but the relations for Wl, 01, M1 and Qa are not so simple in our nonlinear situation. Considering the variations of (6.30) can solve the problem. When the variations of w, 0, M, Q and N are taken as unknowns, the corresponding differential equations and thus their difference equations (the latter are omitted here) are linear. For t  0 these are d(~ dx
= (cos0) k16k0
d(6 k O)
6kM
dx
I+A'o
._ d(6kM)
dx d(6 kQ) dx
(6.32a)
(6.32b)
= 6kQ
(6.32c)
 (q~cosO  q,,sinO) k161'0 + (sinO) k1 6kqs + (cosO) k1 8kqn M k16kN + N k16kM
+
(6.32d)
I+A~/
d(6kN)  (qssinO + qncosO) k1 61'0 dx Mk1 o~Q
(cosO)k16kqs +
(sinO) k1 o~q,,
+ Qk1 o~M (6.32e)
l+AT/ where a variation, for example, is recorded as 6kw  w k

Wk1 , which might be produced
Ch. 6
76
Large Deflection Analysis of Poroelastic Beams
by a change of loading functions, boundary values or whatever. In a practical procedure (6.32) is considered as discretized, where k refers to an imaginary loading step. In each step the changes of the loading are finite so that the given loads can be applied in a finite number of steps. This is similar to what was done previously, except here variations are investigated. Consider now the time change. Rewriting Eq. (6.23) and substituting into (6.24) and (6.25) to replace Off&, the following system of differential equations for t > 0 is found: 0w
= sin0
(6.33a)
O0 3x  r/Mp  M
(6.33b)
OM  Q Ox
(6.33c)
Ox
OQ Ox




qnCOSO + q s s i n O  N(rlM p  M)
(6.33d)
q , , s i n 0  qscosO + Q(rlM p  M)
(6.33e)
 ( l nt /~'O)Mp  ~ ' /
(6.33f)
ON Ox
o3X2
Integrating Eq. (6.33f) with respect to time over [tj1, tj] and using the trapezoid formula to compute the integral of the lefthand side, as well as writing the remaining equations of (6.33) in incremental forms with respect to the time t, yields 09(AjW)
__ COS0 j  1Aj 0
(6.34a)
3x
0(N0) Ox
o(AJM) 0x
o(AJQ) Ox
rlAJMp  AJM
 ~JQ
(6.34b)
(6.34c)
_ (qscosO j1 _ qnsinOJ1)AJ o  (riM p  M)JI AJN  NJI (~qAJMp  AiM) (6.34d)
o(AJN) Ox
 (qssinO j1 + qncosOJ1)AJo + (~TMp  M ~  I A J Q + Q/I(TIAJMp  A J M ) (6.34e)
77
N u m e r i c a l P r o c e d u r e f o r the F i n i t e D i f f e r e n c e M e t h o d
(1 + Ar/)(Mp)/  ~
0x~
 A A J M + (1 + A~7)(Mp)/1 +
2 (6.34f)
where j > 1, and the time increment at the time step j is defined as AJt  tj  tj1. The increments of the functions are recorded as AJw  w/  w/ 1, and so on. It was assumed that the loads are not functions of time. If the loads are timedependent, additional items should be added to the fourth and fifth equations of (6.34). Thus, the solutions for longer times can be obtained successively, i.e. no simultaneity is required for all times. For the spatial coordinate x, when j is given, all equations of (6.34) can be manipulated in the same way as was done previously except for the last one, for which a second order difference for Mp should be introduced at time tj. A 5node central difference formula given as
(
o92Mp ~
~. =
k= 2
i j
(6.35)
C'kMi+ ~
is adopted when 3 ~ i < m  2. The coefficients are given by Eq. (D. 1) in Appendix D. When i : 2 or i = m  1, a 3node central difference formula is employed. The coefficients are shown in Table D.1 of Appendix D. When i = 1 or i = m, the difference equations for (6.340 are replaced by the diffusion boundary conditions in discrete forms. They are ( M p)/= j 0
(6.36)
for permeable boundaries, and/or +
+
0
(6.37)
for an impermeable boundary at the leâ€˘hand end of a beam where the 3node forward formula is used. If the righthand end is impermeable, a similar condition to (6.37) can be obtained by a backward formula. The system of difference equations then obtained for (6.34f) has the same form as that for the remaining equations of (6.34), although the elements of the coefficient matrix (which is also banded with bandwidth being five) take different values. Therefore, Eqs. (6.34) can be solved in a similar way to Eqs. (6.32). The full procedure in solving the nonlinear poroelastic problem is outlined in the following section.
6.4. Numerical Procedure for the Finite Difference Method The numerical procedure for solving the nonlinear poroelastic problem is presented here. There would be no difficulty in extracting solutions for generally timedependent loads and boundary conditions; however, for the sake of simplicity, it is assumed that the loads are imposed at t = 0 and that they are n o t dependent on time thereafter (the loads may of course be functions of x).
78
Ch. 6
Large Deflection Analysis o f Poroelastic Beams
(I) If any of the given boundary values (displacement and/or external load) are not zero, we can imaginarily load the nonzero values in steps and use k to refer to a loading step. At this stage the distributed loads are not to be loaded, i.e. q,,  qs  6kq,, = ~  0 (for t = 0). 1. Let k  0 and take Wk, Ok, M k, Qk and N k to be identically zero (note that all boundary values are zero at this moment)" 2. let k increase its value by 1 and load a set of increments for the given boundary values; 3. determine the nodal values of o~w, o"k 0, 6kM, 64'Q, and o"kNproduced by the increments of the boundary values. First solve successively for o~Q, 6kM, o~ 0 and 6kw, respectively by the corresponding difference equations of (6.32d), (6.32c), (6.32b) and (6.32a). This is performed on the supposition that they are zero at the node xl and then modified by Eq. (E.3) with the constants being determined by the given boundary increments (when k > 1 initial guesses for ffM and 6kN are necessary, and can be taken from the last step, i.e. taken from those for k  1). Then solve for 6"kN by the corresponding difference equations of (6.32e) using the given boundary increment; 4. repeat step 3 until the desired accuracy has been achieved; 5. calculate w k  w k 1 nt ~kw, ok _ ok1 nt ~k o, M k  M k1 + 6~M, Qk __ Q k  1 + 8~Q and N k  N k 1 + 6k N ; 6. if the boundary values have been loaded to the given values, go on to Stage II), otherwise return to step 2 above. (II) If the distributed loads (qn and/or qs) are not zero, they can imaginarily be applied in a stepwise manner, and use k to refer to a loading step; now all given boundary values are taken to be zero (for t = 0). 1. Let k = 0, (q,,)k = 0 and (q,)k = 0; Wk, Ok, M k, Qk and N k a r e taken to be the final results of Stage I, if these exist, or taken to be zero otherwise; 2. let k increase its value by 1 and give b'kq~ and 6"kqs; 3. give an initial guess for 6~0, 6"kM, and 6"kN, which are necessary for solving foro~Q: if k = 1 let them be zero, otherwise let them be equal respectively to those for k  1; 4. determine the nodal values of 64'w, 6"k0, ~ M , o"kQ, and 6"kNproduced by the increments of the loads. First solve successively for o~Q, ~ M , ~ 0, and ~w, respectively by the corresponding difference equations of (6.32d), (6.32c), (6.32b) and (6.32a). This is performed on the supposition that they are zero at the node xl, and then modify them by Eq. (E.3) with the constants being determined by the correct boundary conditions (i.e. the value is taken to be zero at a prescribed boundary). Then solve for 64'N by the corresponding difference equations of (6.32e) using the correct boundary condition; 5. repeat step 4 until the desired accuracy has been achieved; 6. calculate w k = W k1 + i~kw, Ok  0 k1 nt ~ko, M k = M k1 + 6kM, Q k = Qk1 + 6kQ N k = Nk1 + 6kN, (qn) k = (%)~:'+6~qn and (qs) k= (qs)kl+ O~qs; 7. if (q~)k= q~ and (qs) k= qs go to Stage III, otherwise return to step 2 above. (III) Calculate Mp at t  0 by (6.27), where M is now available. (IV) Calculate all nodal values for t > 0. qn and qs take on the given values (which are independent of time). All given boundary values are taken to be zero.
Examples and Discussion
79 9 j
J
1. Let j  0, and w/, 0/, Mj, Qr N and ( M . p ) are taken to be the final results for t = 0; 2. let j increase its value by 1 and give AJt; 3. give an initial guess for Aj 0, AJM and N N which are produced by the increment of time: if j = 1 let them be zero, otherwise let them be equal respectively to those for j  1; 4. solve the corresponding difference equations of (6.34f) for M p using the given diffusion boundary conditions as denoted by (6.36) and/or ( 3 7 ) , etc. and let AJ(Mp) = (Mp)]  (Mp.)j1; 5. determine Nw, N 0, NM, AJQ and AJN. First solve successively for AJQ, AJM, Aj 0 and AJw, respectively by the corresponding difference equations of (6.34d), (6.34c), (6.34b) and (6.34a) on the supposition that they are zero at the node xl, and then modify them by Eq. (E.4) after the constants are determined by the correct boundary conditions. Then solve for AJN by the corresponding difference equations of (6.34e) using the correct boundary condition; 6. repeat steps 4 and 5 until the desired accuracy has been achieved; 7. calculate w/ = w] 1 + Ajw, ~ = ~1 + Aj O, M j = M j 1 + AjM, ~ = ~  1 + Aj Q and N j = N j1 + AJN; 8. if the results for more times are required, go to step 2 otherwise terminate the computation. \
]
6.5. E x a m p l e s a n d D i s c u s s i o n
In the following, various cases are analyzed using the above numerical procedure. The calculations are convergent for these cases when the loading values can be so high that the maximum rotation of a beam reaches approximately 0.7 rad (40~ When the loading is low, the numerical results obtained from the large deflection formulations are found to converge to the corresponding small deflection solutions, for which the closed form solutions are available (Chapter 3). Once again, it is noted that only two material parameters, A and r/, are involved since all variables are normalized.
A cantilever beam subjected to a uniformly distributed load Consider first the deflection of a cantilever. The conditions for Fig. 6.2a,b are the same except for diffusion boundaries. The load is suddenly applied, constant thereafter, and uniformly distributed. We see that the deflection can be both quantitatively and qualitatively changed if the flow process is changed by the diffusion boundaries. At t = 0 the deflections are the same; these are the instantaneous elastic responses to the load application. At t  0.85 the final shape of the beam has essentially been achieved in each case. One observes different 'creep' rates for the two cases. Note that for t  0.08 the deflection for the permeable case is quite close to the final deflection, while with the beam ends sealed the deflection is closer to that at t = 0 than it is to the final deflection. Furthermore, the rotation of the beam axis for the 'open' beam (the ends are permeable) is monotonic with respect to the position due to quick dissipation of the pore pressure. This is not the case for the sealed beam (i.e. both ends are impermeable).
Ch. 6 Large Deflection Analysis of Poroelastic Beams
80 0.0
o
a)
0.3
o t=0.08
\
~ t=0.85 0.6
.0
.
.
.
.
I
0.5
.
.
.
. 1.0
0.0
0.3
0.6 0.0
0.5
1.0
Horizontal position Fig. 6.2. Shape of a cantilever subjected to a normal uniformly distributed load qn ( = 6) at times t = 0, 0.08 and 0.85 for different diffusion boundaries: (a) permeable at both ends; (b) impermeable at both ends (A = r / = 1).
Fig. 6.3 shows the variation of the beam rotation (0) at some positions along the sealed beam. The pore pressure does not finally vanish in this case. The long time pore pressure Mp is a constant, and it is known from Eq. (6.23) that it produces a constant curvature with the opposite sign to the one produced by the applied load. Consider next the pore pressure distribution for the sealed beam (Fig. 6.4b). At t = 1.00 the pore pressure is essentially constant and equal to 0.440, the value for infinite time. Eq. (2.52b) determines this long time pore pressure, which is valid for both the small and the large deflection theories (when e0 is taken as zero). If the small deflection theory were applied to the case corresponding to Fig. 6.4b, we would find Mp(x, oo) = 1/2. On the other hand, for both ends permeable the initial (t = 0) pore pressure distribution, which is identical to that for the impermeable case, undergoes rapid changes at very short times near the fixed end (see Fig. 6.4a) and of course finally decays to zero. Fig. 6.5 shows results for different loading values and 'mixed' permeability conditions, i.e. impermeable at the wall and permeable at the free end. Fig. 6.5a is the immediate response (t  0) while Fig. 6.5b is for time t = 0.26. Noting that the curves for t  0 refer to elastic solutions when the fluid is trapped, we see for this specific case that when the
Examples and Discussion
81
0.42
4
0.47
nz=0.60 0:=:=0.80 Az=0.95 +x=l.O0
i
0.52
0.57
'
0.00
'
'
'
I
'
0.25
'
'
'
I
.
0.50
.
.
.
i
.
0.75
.
.
.
1.00
t Fig. 6.3. Rotation of beam axis vs. time for a cantilever. Conditions are as for Fig. 6.2b.
1.5
1.5 ~)
1.o1 q
\
mr=0
oto.o8 ~t=0.z8
\
I
I
0.5
1,0
0.5
0.0
0.0 0.0
0.5 X
1.0
~r 0.0
0.5
1.0
27
Fig. 6.4. Pore pressure distributions of a cantilever with different diffusion boundaries, (a) and (b). Conditions are as for Fig. 6.2a,b, respectively.
Ch. 6
82
Large Deflection Analysis of Poroelastic Beams
0.0
0.0 I " ' 1 ~
mq=l oq=4 aq=6
\ ~
0.2
0.2
o
I
0.4
0.4
d
0.6
0.6
0.0
0.5
1.0
0.0
0.5
27
1.0
27
Fig. 6.5. Rotation of beam axis vs. position at times (a) t = 0 and (b) t = 0.26 for a cantilever subjected to a load qn = q ( 1, 4 and 6, respectively) and impermeable at x = 0 and permeable at x = 1 (A = r / = 1).
l o a d i n g is s m a l l , the fluid flow d o e s n o t q u a l i t a t i v e l y c h a n g e the r o t a t i o n patterns. H o w e v e r , for a h i g h l o a d i n g , the p a t t e r n s will be c h a n g e d s o o n after the flow b e g i n s ; n o t e that h e r e t 
0.26 is a s m a l l t i m e c o m p a r e d to t 
c o m p l e t e (at that t i m e , the m a x i m u m r o t a t i o n for q 0.0 t
0.2 
A t the in
0.0
\\"\,\. ~ , , ,
0.2
aq=6
0.4 
0.6
3 w h e n the d i f f u s i o n is n e a r l y
6 is a b o u t 0.8 rad, or 45~
0.4
t
0.0
0.5 27
1.0
081
0.0
~
'
~
'
I
0.5
'
1.0
27
Fig. 6.6. Rotation of beam axis vs. position at times (a) t  0.10 and (b) t  0.26 for a cantilever. Dashed curves refer to those for t  0. Conditions are as for Fig. 6.5 except here the both ends are impermeable.
Examples and Discussion
83
b e t w e e n time, the r o t a t i o n is not m o n o t o n i c e v e n t h o u g h that at l o n g t i m e s m o n o t o n i c i t y m u s t n e c e s s a r i l y exist. O n the o t h e r hand, the i n c r e a s e o f l o a d i n g o n l y q u a n t i t a t i v e l y i n c r e a s e s the r o t a t i o n o f the b e a m axis in an elastic situation. T h e situation is m o r e c o m p l e x in the p o r o e l a s t i c case; w e see that the c u r v a t u r e (=O0/Ox) c h a n g e s its sign a l o n g x w h e n the l o a d i n g is sufficiently high. This is m o r e significant in the case o f the s e a l e d b e a m s h o w n in Fig. 6.6. H e r e the d a s h e d c u r v e s are for t = 0 for the v a r i o u s l o a d i n g
0.0
0.2
a)
0 ....4
0.4
o t=0.42
~"N~
,,t=2.40 + t=3.20
0.6
'
l
0.0
\~~. '
0.2
l
'
0.4
l
0.6
'
"'"
0.8
0.0
o =
0.2
b) Q)
[] t=O o t=0.20
0.4
0.6
t=l.O0 + t=l.20
'
0.0
I
0.2
'
I
0.4
I
0.6
0.8
Horizontal position Fig. 6.7. Shape of a cantilever subjected to a lateral displacement at the free end, which is suddenly applied and remains unchanged thereafter (w(1, t) = A = 0.6). The fixed end (x = 0) is impermeable, A  6 and r / = 1. (a) Permeable at x = 1, the curve for t = 3.20 is overlapped by the one for t = 2.40; (b) impermeable at x  1, the curve for t = 1.00 and 1.20 essentially overlap.
Ch. 6
84
Large Deflection Analysis of Poroelastic Beams
values, and are the same as shown in Fig. 6.5a since they are elastic responses. It is interesting that the rotation at the free end changes very little from the initial elastic response, while the rotation at the middle positions changes significantly.
A cantilever beam subjected to an i n s t a n t a n e o u s deflection Consider now a cantilever, which is given an instantaneous vertical deflection by applying a lateral displacement A at the free end. Now restrain the free end (x = 1) from further vertical movement, i.e. let w(1, t) = A. We observe the shape of the beam as a function of time. Fig. 6.7 shows the cases for different diffusion boundaries. In the case of a beam sealed at both ends (Fig. 6.7b), the vertical deflection grows monotonically except at very near to the 'free' end. If the beam permits diffusion at the 'free' end (Fig. 6.7a), the middle part of the beam reaches its maximum deflection at approximately t   0 . 4 2 . The deflection here then decays and the beam approaches its final shape at t = 2.40. This final shape, however, is not the initial shape as would be the case for small deflection. To further clarify the situation we focus on the change of the vertical displacement w at the center of the beam, for situations when A takes different values; the beam is permeable at the free end. Fig. 6.8 shows that if the deflection is very small, the beam will finally
0.0156
0.062 t
0.0178
0.072
0.13
0.20
0.15
0.24 0
1
2
t
3
4
0
1
2
3
4
t
Fig. 6.8. Lateral displacement vs. time at x  0.5 of a beam as in Fig. 6.7a with A taking different values.
Examples and Discussion
85
0.0 or=0.03
/
or:0.20
/ / / ~
0"0 I o x = O . O
//I
a~~"~_
J
0.2"
0.2
O.4
0.4
0.6 0.5
1
0
z
1
2
t
Fig. 6.9. Pore pressure for a simplysupportedbeam subjected to a normal uniformly distributed load qn ( 10) with boundaries impermeableat x  0 and permeable at x  1 for A  77= 1: (a) pore pressure distributions at times t  0.03, 0.20 and 1.00; (b) pore pressure decay at positions x  0.0, 0.3 and 0.7. return to its initial elastic position. In a large deflection situation, the beam cannot return to the elastic position after diffusion is finished. The larger the deflection, the farther (relatively) the final position is from the elastic position.
A simply supported beam subjected to a uniformly distributed load Consider now the pore pressure for a simply supported beam (one end is free to move horizontally) subjected to a uniformly distributed load. One end of the beam is impermeable and the other is permeable. For such diffusion boundaries the diffusion progresses slower than it would with the same diffusion boundaries at both ends. Pore pressures are shown in Fig. 6.9. For this beam, the fluid flow is not negligible until about t = 5, although the pore pressure has been reduced to less than 20% of its peak value by t = 2. The initial state is such that M and thus Mp are symmetric with respect to x = 0.5 (necessarily zero at both ends). One observes that the pore pressure at the sealed end (x = 0) increases very quickly soon after the diffusion begins and the symmetry is broken. After it reaches the peak value at about t = 0.20, it decays and finally vanishes.
A clamped beam subjected to a uniformly distributed load Figs. 6.10 and 6.11 show the results for a beam fixed and impermeable at the both ends (one end is free to move horizontally), subjected to a uniform load. In Fig. 6.10, exactly antisymmetric patterns for the rotation are observed as expected. Fig. 6.11 illustrates the way the pore pressure vanishes in such a beam. (Note that the pore pressure finally vanishes even though the beam is impermeable at both ends; cf. Section 3.2.)
Ch. 6
86 0.70
Large Deflection Analysis of Poroelastic Beams

( I
0.35
U
0.00
0.35
m t=0 o t=0.04 a t=0.26
0.70

0.0
0.2
0.4
0.6
0.8


!
1.0
Fig. 6.10. Rotation of beam axis vs. position for a beam clamped and impermeable at both ends A = r / = 1).
(qn

100,
4.0 t=O o t=O.01 A t=O.08 + t=0.26 o
,
2.0
0.0
2.0
~
0.0
0.1
0.2
I ~
0.3
0.4
0.5
Fig. 6.11. Pore pressure distributions of a beam clamped and impermeable at both ends (q, = 100, A = 77 = 1). Half is shown due to symmetry.
Examples and Discussion
87
0.2
0
0.0
~
0.2
0.4
3 :[ ~
0.6
,
0
1
t
2
4 I 0
o Small deflection theory 0 Large deflection theory
. . . .
,
1
. . . . 2
t
Fig. 6.12. Pore pressure decay at positions (a) x = 0.27 and (b) x  0.60 for a beam which is fixed and impermeable at x = 0 and simply supported and permeable at x = 1. (qn  25), A = 4, 7/= 0.25.
A beam clamped at one end and simply supported at the other, subjected to a uniformly distributed load Finally, consider a beam clamped at one end and simply supported at the other, subjected to a uniformly distributed load. The differences between the solutions obtained respectively by the small and large deflection theories, when the loading is very high (i.e. the deflection produced is very large) are shown in Fig. 6.12. A and ~/have been assigned values for which the differences are easy to observe. Here one clearly sees the necessity of employing a large deflection theory. The pore pressure moment, which finally decays in the beam, can suddenly increase at certain positions for a period of time. This is seen dramatically in Fig. 6.12a for a position on the beam (x = 0.27) where the pore pressure is generally relatively low. The sign of the pressure even changes two times (cf. Fig. 4.3). A much sharper change is observed in the large deflection situation than for the small deflection approximation. For locations on the beam where pore pressure is relatively high (x = 0.60) we see from Fig. 6.12b that at short times there are still significant differences between small and large deflection theories. In summary, it has been shown that the numerical technique developed here makes it possible to determine deflections and pore pressures for beamlike elements made of a poroelastic material exhibiting axial fluid diffusion when subjected to general loading and having any boundary and diffusion conditions.
This Page Intentionally Left Blank
Chapter 7 STABILITY OF POROELASTIC COLUMNS
In this chapter, we consider three stability problems of poroelastic columns: quasistatic buckling, postbuckling and dynamic stability. For the buckling problem, the timedependent behaviors of the critical loads and deflections are considered for various diffusion and geometrical boundaries. The timedependent column response for the case of a constant load is also given. For the postbuckling problem, the timedependent behavior of the columns, governed by three coupled equations, is obtained by using the large deflection theory. These equations are transformed into a single one, enabling the analytical derivation of the initial and the final responses. It is shown that, unlike the quasistatic response obtained by using the small deflection theory, the long time response derived here is bounded. The imperfection sensitivity of these columns is also investigated. For the dynamic stability problem, the stability conditions and boundaries are found, together with the critical (minimum) loading amplitude for which instability may occur.
7.1. B u c k l i n g o f C o l u m n s
Fluidsaturated poroelastic beams subjected to axial loads P are shown in Fig. 7.1. w denotes the transverse deflection of the beam axis if buckling occurs. Again, only nondimensional variables are considered. Quantities are normalized as in Eqs. (2.44). In addition, the point load is normalized as follows (7.1)
P* = PLZ/EI
The superscript * will be omitted hereafter for simplicity. The partial differential equations governing the problem are 0x2
32Mp o~X2
(2.45b)
 ~'/Mp + M = 0
3Mp 
Ot
33w  A
o~X2 0t
 0
(2.48b)
Among the three unknowns, w, Mp and M, the last can be determined in terms of the first by equilibrium of the beam in an obvious manner for the situation of statically determinate problems. For a simply supported beam, M = Pw.
90
Ch. 7
Stability of Poroelastic Columns
w P
x
(a)
(b)
Fig. 7.1. Beams subjected to axial loads: (a) a simply supported beam; (b) a cantilever; (c) a beam clamped at x  0 and hinged at x  L.
The geometrical boundary conditions for w are the same as those in the elastic case. The diffusion boundary conditions imply conditions on w. For a free end, for instance, if no transverse loads are applied at the end, then considering Eq. (2.45b), the permeable diffusion boundary condition is given by W, xx = 0 and the impermeable diffusion boundary condition is given by W,xxx  0. After Mp is eliminated from the two governing equations, (2.45b) and (2.48b), one obtains 04w o~X4
o4M +
o~X2
33w = (1 + An)
o~X2 3t
3/14 +
3t
The required initial condition when a load is suddenly applied at t   0 obtained from Eq. (2.49b), by inserting (2.27b). Then we have 02w 3x 2
(7.2) is readily
m  
~
1 + h'q'
t:
0
(7.3)
7.2. L i m i t s o f C r i t i c a l L o a d
The buckling of the poroelastic beam is generally expected to be a timedependent process due to the viscosity of the pore fluid. Three ranges of loading must be considered: for t  0, for t = oo and for any t in between. The lower and upper limits of the middle range will now be determined.
Limits of Critical Load
91
Using PcE to represent the dimensionless critical load for the poroelastic beam with no pore pressure, i.e. the drained elastic solution, it is concluded from Eq. (7.3) that the critical load for the case in which the fluid is trapped is (1 + Ar/)PcE. This is the maximum critical load for the poroelastic beam because a load which is greater than this value will result in purely elastic buckling before any diffusion has time to take place (Biot, 1964). Thus, pmax E cr = (1 + Ar/)Pcr
(7.4)
Further, it is noted that the critical load for the poroelastic beam, Pcr, should not be less than PcE. If the beam is not sealed at both ends, then when P  PcE is applied and the pore fluid can flow out of the beam (due to even the tiniest imperfection), albeit slowly, the pore pressure would decrease. After an infinite time, the beam will reach its balance in an elastic state (i.e. there will be no fluid flow and no pore pressure at all). Thus the minimum possible critical load for a beam with at least one end permeable is pmin __ P~cr
(7.5)
Consider now the lower limit for a sealed beam, i.e. the beam is impermeable at the both ends. After being integrated over [0,1] with respect to x, and with the impermeable diffusion boundary condition (i.e. Mp,x0) applied to both ends of the beam, Eq. (2.48b) can be written as follows:
0 10( w) Mp +
d
0
(7.6)
where it has been assumed that the operations of derivative with respect to t and the integration can be interchanged. Thus the integral in Eq. (7.6) is not a function of t but is rather a constant. Further, that this constant should be zero, is implied by Eqs. (2.45b) and (7.3) at t = 0 when the fluid is trapped. Actually, the conclusion that this constant is zero does not depend on the initial conditions. It is a specific form of Eq. (2.41b), after M is replaced by (2.14b). Hence Mp+A~x 2
dx=0
(7.7)
This is true for all times. As a special case consider t = oo, when Mp(x, oo) should be a constant with respect to position since no diffusion occurs then. Then from Eq. (7.7), 
A
[ow

ow
]
(7.8)
As mentioned above, if Mp  0 then the lower limit is given by Eq. (7.5). Therefore, Eq. (7.5) can also apply to a sealed beam (i.e. both ends impermeable), as long as the requirement below is satisfied 0w 0w ~x (0, t)  ~x (1' t)
(7.9)
This is the case for the beams shown in Fig. 7.2a,b. For cases in which Eq. (7.9) is not
Ch. 7
92
Stability of Poroelastic Columns
(a)
(b) Fig. 7.2. B e a m s with rotation being zero at both ends.
justified Mp depends on w, which is not a priori determinable in a buckling situation. Thus the lower limit cannot be determined in an obvious manner, but clearly crPmin~ PerE. 7.3. T i m e  D e p e n d e n c e of Critical Load and Deflections
The timedependent nature of the critical load will now be investigated. We first define what is meant by a timedependent critical load. Suppose that a load P0 (pcmrin ~ P0 < pcr m a x ]J has been applied to the beam at t __ 0 and the deflection is w(x, O) __ ~(x) at this moment; we now determine the time dependent change of the critical load Pcr(t) when the deflection is required to remain unchanged with time (i.e. w(x, t) = r t > 0). The point is that if at any time P(t) > Pcr(t) the deflection will increase, leading ultimately to buckling. Then, by Eq. (7.2), we arrive at d4~
oZm
t
dx 4

o~X2
om
(7.10)
o~t
M can be obtained by the equilibrium of the beams. For the beams shown in Fig. 7.1, M can be expressed as follows:
M
P~
Fig. 7.1a
P ( v ~  6)
Fig. 7.1b
P~R(1x)
Fig. 7.1c
(7.11)
where 6 is a constant and the reaction R is a function of time. Thus Eq. (7.10) can be written as d4~ d2~, dx 4 + P  ~ (~
dP ~b) d7
(7.12)
where (h, in accordance with the different cases, is defined as
~b
0
Fig. 7.1a
6
Fig. 7.1b
Ro(1  x)
Fig. 7.1c
('7.13)
TimeDependence of Critical Load and Deflections
93
For the beam shown in Fig. 7. lc, the following relationship has been used:
R(t) = RoP(t)
(7.14)
where a dot refers to a derivative with respect to t and R0 is a constant. If the relation were not true, the fighthand side of Eq. (7.12) would be ~(x)P(t)  (1  x)[~(t). Then a particular solution of Eq. (7.12) for v~(x) would be (1  x)[~(t)/P(t), which would be a function of t. This is in contradiction to the definition of v~(x). So Eq. (7.14) is justified. Actually Eq. (7.12) can be adapted to all types of beams, with q5 being a constant or a linear function of x. Now, solving Eq. (7.12) for P(t) yields
P(t)
l~(4)(X) +
P0 +
~"(x)
exp
~"(x)
t
(7 15)
~(x)  4,
where a prime refers to a derivative with respect to x. Since the fighthand side of the above expression is not a function of x, one concludes that ~(4)(X )
~"(x)
~tt(X)
  C l ,
v~(x) q5
c2
(7.16)
where Cl and c2 are constants. Further, the two constants should be equal so that the solution can exist (~b is at most linear). Also it is clear that the constants are positive. Then c l = c2   c 2 can be determined by solving the boundary problems. However, its value can be determined by noting that p(oo)  c 2 and thus c 2 _ lcrPmin W e note that c 2 can take on higher values also, which correspond to higher mode shapes. By taking c 2 to have the unique value pmin cr we then limit v~(x). But for our purposes this is satisfactory, since the higher c 2 imply much greater elastic buckling loads and our purpose here is only to determine load paths guaranteed not to cause buckling (growth of deflection). Hence the critical load is given by Per(t) = cr pmin + (P0  Pcr min)
e P e rmin., x t) p
(
(
7
.
1
7
)
We can also determine the buckling modes by Eq. (7.16) and the corresponding geometric boundary conditions, giving as follows for the different cases shown in Fig. 7.1:
v~(x) =
8 sin(cx)
for Fig. 7.1 a, c = n 7r
811  cos(cx)]
for Fig. 7.1b, c  (2n  1 ) rr/2
Ro[COS(CX) + sin(cx)/c + 1  x]
for Fig. 7.1c, c = tan(c)
(7.18)
where n  1, 2, 3 ..... and only n = 1 is physically viable if there is no constraint between the ends of a beam. v~(x) is found to satisfy the permeable diffusion boundary conditions at both ends for the case shown in Fig. 7.1a, and the impermeable diffusion boundary condition at x = 0 and permeable diffusion boundary condition at x= 1 for the cases shown in Figs. 7.1b and 7.1c. Thus, these diffusion boundary conditions are required in order that Eq. (7.17) is justified for the beams shown in Fig. 7.1. Generally speaking, the requirement is that the pore pressure vanishes at long times, or that pmin __ pcE. In the limiting situation, when P0 takes the value pmax _ (1 + Ar/)PcEr, the beam buckles instantaneously. Then Eq. (7.17) takes its special form (just below pmax)
94
Ch. 7
Stability of Poroelastic Columns
Pcr(/) = pcEr[1 + /~'17 exp(PcErt)]
(7.19)
On the other hand, if P0 takes the value Pc~r, it will take an infinite time before the beam buckles. At this point it is necessary to reconsider Eq. (7.17) and to clearly understand its implications. First, recall that a critical situation is defined as that when an initial existing deviation of the column axis from straightness  imposed by any means whatsoever will remain constant. Then Eq. (7.17) prescribes a loading path for a load whose initial value is P0 so that a critical situation is achieved. Any deviation, which brings the loading path above that given by Eq. (7.17), will ultimately lead to buckling. Now suppose the loading path up to some time t was such that it did not result in a critical situation. Then, what continued loading for time t + r (r > 0) will result in a critical situation? This is answered by considering Eq. (7.12) and noting that its solution was not in any way dependent on the past history (prior to r = 0+) of the column. Thus the following expression is found to be fully equivalent to Eq. (7.17): Pcr(t + r) = pmin + [ P ( t )  pmin]exp(pcmrinr)
(7.20)
In other words, Eq. (7.20) is identical in form to Eq. (7.17) with P0 replaced by P(t), the 'current' value of the load at time t which does not cause buckling. Again, any deviation, even for the shortest time, which brings the loading path above that given by Eq. (7.20), can lead to buckling. In other words to avoid buckling the slope must be more negative than that of the expression in Eq. (7.20). Taking the derivative of Eq. (7.20) with respect to r and setting r = 0+, the requirement dt 0 is required so that there will be one real root and two conjugate complex roots as given below: ~1  
1
 2c~,
where i  x/S 1 and
~2  c~ + i~,
~:3  c~  i ~
(8.87)
Ch. 8 Analysis of Poroelastic Plates
130
c~~1 (~/g//2 +
1
'V~  (  g / / 2 + V/A)  ~
(8.88) are real quantities. Therefore the general solution in real form can be written as "" ff)ij __ exp(aij6t) [ C(exp(aij~ot) + C'~~jCOS(OgijJ3t)
+ C~sin(a/jB t)]
(8.89)
where ~:o = ~ 1  6 =  1  3 6
(8.90)
Given the initial values as in Eq. (8.82), the constants C~ are found below: Cf = ((~2 _jr_~2)0/2(])ij(0)_
C~  r
c~ 

2~l.Olij~ij(O) + ~ij(O)
C~
(8.91)
d~o(o) _ a +ij(o)_+ ~oc[ /3~ij /3
Two types of initial pore pressure are investigated: (I) no initial pore pressure; and (II) the initial pore pressure is prescribed by Eq. (8.42), i.e. when after displacement the plate is released before the pore pressure has had time to decay at all. The initial deflection considered is w  sin(Trx)sin(,rry) and the initial speed w is taken to be zero. 1
1
c=5
tc=l
~0
I
0
II 

1
t
2
0.0
0.1

!


0.2
t
Fig. 8.9. Decay of the deflection amplitude of a rectangular plate for At/= 0.25 and 3' = 1.5 in free vibrations for different initial pore pressure.
Transverse Vibrations of Simply Supported Plates
131
Some examples of free vibrations are shown in Figs. 8.9 and 8.10. As expected, the frequencies increase when K or At/increase (or y decreases which is not shown). In the cases shown in Fig. 8.9, the differences between the curves for initial conditions Types I and II are not large if other conditions are the same. However, Fig. 8.10 shows that the differences can be significant (cf. Fig. 5.4). It is clear that the natural frequencies and At/ make the difference: the higher the natural frequency, the bigger the differences are between the curves for the two types of initial pore pressure if At/ is sufficiently large. The phenomenon is explained here from the point of view of energy loss, which depends both on the actual fluid speed and on the comprehensive poroelastic characteristics represented by A~/. When the natural frequency is low, the fluid flows slowly so that little energy is lost in a cycle due to viscosity of the pore fluid. Therefore the behavior of the system approaches that of an elastic system and the existence of the initial pore pressure is not important. The same is true for a very small A~/. In the opposite situation, the energy dissipation is significant when the fluid flows quickly so that the initial pore pressure plays an important role in the process. When no initial pore pressure exists (Type I), there will be no initial impetus for fluid movement (i.e. there are no pressure gradients). The fluid flow, which is then produced by the deflection change, is not able to achieve a high velocity in a short time due to high damping. Hence the plate can only move slowly at first. In other words, the potential energy, which is produced by the initial deflection, is not able to be convened predominantly into kinetic energy but takes the path of being dissipated finally before the fluid reaches its high velocity. On the other hand, when there exist large pore pressure gradients at the beginning, the fluid flow can attain high velocity quickly and thus the plate vibrates. The energy is converting quickly among the following: the potential energy of the plate, the potential energy of the fluid produced by the pore pressure gradients and the kinetic energy of the plate. The energy loss occurs much quicker than for Type I due to the rapid fluid flow. 1
1
~r1=0.75
0
I
aa
] I 0
. . . .
tion I ~176176 II , . . . . , 1 2
t
0
I 0
, 0.5
, I
t
Fig. 8.10. Decay of the deflection amplitudeof a square plate for Y= 1 in free vibrations for different initial pore pressure.
Ch. 8
132
Analysis of Poroelastic Plates
Forced vibrations The long time solutions for harmonic loading problems of the same plate, q(x],x2, t) ~(Xl ,x2)exp(icot) can also be extracted. For this purpose, the coefficients in Eqs. (8.79) and (8.80) can be written in the forms
dPij(t)  wijexp(icot),
Bij(t) : bijexp(icot)
(8.92)
Then Eq. (8.83) becomes an algebraic equation for the given i and j and thus wij is determined
(c~ij + iw)bij wij :
(8.93)
a/~ + i(1 + A~/)c~2co yzc~ijco2  i y 2 c o 3
The first two resonance regions are shown in Fig. 8.11, where co is the circular frequency of the loading. For co = 0, the long time solutions for the corresponding quasistatic problems ( : elastic static solutions) is derived. A~ : 0 corresponds to the elastic cases. It can be seen that: (a) the resonance areas are shifted to the fight; (b) these shifts are more distinguishable for higher frequency; and (c) the amplitude response is reduced when A ~ increases. Fig. 8.12 shows the situation for plates with different ratio of length and width. Since L1 is kept constant, larger K refers to a narrower plate, for which the resonance areas are at higher frequencies and the amplitude responses are smaller, with respects to the counterpart results of the wider plate. The same behavior is observed also in Fig. 8.13, showing the influence of K, A~/ and y for a given frequency. Note that higher
10~
v ~,rl=O. o ),~=0.1
10'
9" 
.
=lOo~,,i
10 3
10 4
.
0.0
.
.
.
'
I
6.0
.
.
.
.
.
I
.
2.o
.
.
.
.
I
.
.
.
.
18.o
Fig. 8.11. Resonance areas for a square plate for y = 0.75.
'
i
z4.o
Transverse Vibrations of Simply Supported Plates
133
lo o

o R:=I
;k~=O
x R:=I
X~=O
o~=l.
~,~/=0.
"0
",_.~
4
/
10.3 0.0
'
'
I
3.0
'
'
f CO/2~r
Fig. 8.12. First resonance areas for plates with K(=
l~176 I
I
6.0
L1/L2)=
'
'
I
9.0
1.0 and 1.5 for y = 0.75.
o,:l.o x,7=o.o o R;=I.O ~,~=0.3
1 0 ~
#,
~=1.5
x~=0.0
I
+ ~:=1.5 X~=0.3
1 0 2
1 0 3
1 0 4
, 0.0
2.0
, 4.0
6.0
Fig. 8.13. Resonance areas with respect to y for f = 2.5.
8.0
Ch. 8 Analysis of Poroelastic Plates
134
values of y denote higher permeability, when the other characteristics of the plate remain unchanged. Finally, as discussed in the beam case, the time scales of the present system are considered. Since the geometrical parameter K is involved, these time scales must be introduced as depending on the bending mode, denoted by positive integers m and n. The characteristic diffusion time is found to be
7/nDD n = (m 2 + K2n2)Tr2K
(8.94)
which refers to the case when after an initially imposed deflection the plate is restrained from further deformation after the initial pore pressure is produced. If the plate is free to deform when a sinusoidal load is suddenly applied the characteristic diffusion times are given by 7/flFn = (1 + /~)7/~D n
(8.95)
The characteristic times for the drained elastic plate are defined as the inverse of the natural (circular) frequency of a simply supported rectangular plate (elastic)
7~sn (m 2 + K2n2)Tr2
/~I
(8.96)
Accordingly, the characteristic times for the poroelastic plate when the fluid are trapped is introduced as follows" '~Tn =
~S n
(8.97)
+an This is obtained by comparing Eq. (8.36) with the parallel equation for the elastic plate. Note that as in the beam case, here too only three independent time scales exist. Namely, the time dependent features in terms of characteristic times are the same as in the beam case, though the ratio of length and width of the plate is necessarily involved.
Chapter 9 CLOSURE
In the previous chapters we have considered poroelastic structures for which fluid diffusion is possible only in the axial (or inplane) directions. After formulating the model we presented solution procedures which made it possible to determine the response of various structures under different loading conditions. Many examples have been presented. Throughout, we have seen response patterns which are unique to this type of element; they are unforeseen, indeed surprising, in terms of experience with structures whose constituting material evinces the most common type of timedependent behavior, viz. viscoelasticity. What makes such a system unique is that to a certain extent the right hand does know what the left hand is doing. Because of the axial fluid flow, information is being passed along the length of an element; thus the time dependent behavior of such structures shows anomalies which cannot appear in viscoelastic structures. This reality, together with the fact that these timedependent behaviors are qualitatively changed when permeability end conditions are altered, makes such structures promising for use in a 'smart structure' environment; these end conditions should be readily controllable at any time based on prior response of the structure. Indeed the permeability conditions could be imposed, or altered, at any position along the beam; this broadens the range of control possibilities. (Such an example is given in Li et al. (1999a); there, an arbitrary value of Mp is imposed at a given point along the beam.) In addition, it is not necessary that the crosssection, or even the properties of the pore fluid, be uniform along the length of the beam. It also should be recalled that the viscosity of many fluids is very sensitive to environmental variables, most especially temperature, and thus characteristic times of the structure could be keyed to the environment. Human technology has not by choice utilized poroelastic materials up to now. Both geological structures and biological materials displaying poroelastic behavior have been imposed upon us; their behaviors have indeed been analyzed. It should be possible to design with poroelastic materials of the particular class considered here to achieve controllable structures. Now we underscore another interesting feature of such a poroelastic material when used in a structure. Under suddenly applied loads the stress experienced by the skeletal material may be significantly reduced, visavis the stress due to slowly applied l o a d s  especially in a structure with at least one permeable end condition. This is so since immediately after the load application a pore pressure is developed (see Eq. 2.13), and this pore pressure bears a portion of the load. Combining Eqs. (2.13) and (2.1) we have that the initial partial stress acting in the skeletal material at any point is given by
136
Ch. 9
crx 
(
1
)
A 1 +  ~ q5 ~'~
Closure
(9.1)
We recall that 4, is the pore volume fraction. However, at long times o'x = %,. Thus a load applied for a very short time with respect to the characteristic time of the structure will cause greatly reduced stress in the skeletal material. The abovementioned technological possibilities are brought home by considering that all of the analyses of structures in the previous chapters (except Chapter 8) are valid for beamlike elements which have geometrically discrete crosssections, i.e. the crosssection is not treated as a single continuum. One can construct a beam crosssection consisting of two tubes, fluid filled, in which are permeable partitions (Fig. 9.1). Even the simple example of a cantilever beam with freeend valves subjected to tip loading as shown in Fig. 9.2 immediately stimulates speculations as to possible control schemes. We show the total analogy as follows. Consider the beam element shown in Fig. 9.3. For simplicity we limit ourselves to bending only (no axial load) and to a section symmetric about the bending axis; it is not necessary that the tube crosssection be round. The bending moment at the section is M; the fluid pressure in the lower cavity is pf. Relying on classical beam theory we can now immediately write 1 r
  S l l M + S12Pf,
2
dVsf  S 2 1 M q S22Pf dx
(9.2a, b)
Here r is the radius of curvature of the beam axis and dVs_f/dx is the change per unit length of the difference between the cavity volume and the fluid volume for the lower half. It is clear that $11  1~El, where E is the Young' s modulus of the s o l i d m a t e r i a l (taken here to
Fig. 9.1. Segment of discrete model of poroelastic beam.
Closure
137
Fig. 9.2. Cantilever beam.
Fig. 9.3. Beam slice.
138
Ch. 9
Closure
be isotropic) and I is the moment of inertia of the crosssection (excluding the cavities). It can be shown that the S matrix is symmetric, i.e. $12  $21. $12 and $22 can be calculated. $12 is dependent only on the crosssection shape and on the elastic properties of the solid material; $22 could be written as $22  S~2 nt
2A
(9.3)
B
S~2 is also dependent only on the crosssection shape and on the elastic properties of the solid material; B is the bulk stiffness of the fluid and A is the crosssectional area of the tube. For instance for the crosssection shown in Fig. 9.1 we find 1 _ 27rhRE[R 2 + 2b 2], Sll
"n'R3 ( $22 =   ~
5
4/,'
S12 = ~
2 
1
1 + (1/2)(R/b) 2
] (9.4a, b, c)
(121') 2 ) "rrR2 1 + 2(b/R) 2 + 2    ~
where it has been assumed that the solid material is isotropic (at least in the tube wall surface) and v is the Poisson's ratio of the material; we have taken h > R and is given by AD =
1 4 + 2(hE/RB)
(9.12)
So even for a very incompressible fluid, or relatively very thin tube wall,/~D could not be greater than 0.25. The product /~D'oD then could approach 0.25. This represents a not insignificant effect. As Poisson' s ratios increase towards 0.5, the limit for isotropic materials, the product decreases. Eq. (9.12) shows the great influence that the relative stiffness of the fluid with respect to the solid and the relative area of the fluid in the cross section have on determining the magnitude of the poroelastic effect; this is as expected. If the tube wall is not isotropic (say is orthotropic) far higher values could easily be designed into the system by properly adjusting the relative longitudinal and circumferential elastic properties.
140
Ch. 9
Closure
Table 9.1. Values of AD and /]D for the simple model /~
77D
0
1
0.25
0.5
0.5
0
0.75
 0.5
1
1
h D (b >> R)
hD
(b = R)
1
1
4 + 2(hE/RB)
6 + 3(hE/RB)
0.5
0.5
3.75 + 2(hE/RB)
5.625 + 3(hE/RB)
0
0 0.5
0.5
1.75 + 2(hE/RB) 1
2.625 + 3(hE/RB) 1
2(hE/RB)
3(hE/RB)
We now illustrate a very unique case demonstrating another peculiarity first noted in Section 2.1 for the continuum case. Consider a filament wound tube with + 45 ~ winding (say glass or graphite fibers in a polymeric matrix). We can use Eqs. (9.11) without having to generalize the S values given in Eqs. (9.4), for the isotropic case. This is so since such a material, in the tube's longitudinal and circumferential directions, has the same properties; thus Eqs. (9.4) and hence Eqs. (9.11) are valid as shown. Also it is known that such a material (in the tube surface, which is the operative situation visavis the elastic constants) has longitudinal and circumferential Poisson's ratios which can approach 1. If v is greater than 0.5 then both AD and ~/D are negative. This was pointed out in Section 2.1 to be physically possible for the continuum case. Examination of Eqs. (9.11) immediately shows the great sensitivity of the extent of the poroelastic effect to both the material and geometric parameters. As v approaches 1 very large values of the product AD~/D can be attained. Table 9.1 shows the values of AD and r/D for various Poisson's ratios and geometries within the very simple model considered. Finally we point out again that tube crosssections need not be circular; the crosssectional shape would have a dominant impact on the values of the two parameters.
REFERENCES Abousleiman, Y., Cheng, A. H. D., Cui, L., Detournay, E. and Roegiers, J. C. (1996). Mandel' s problem revisited. Geotechnique, 46, 187195. Almeida, E. S. and Spilker, R. L. (1998). Finite element formulations for hyperelastic transversely isotropic biphasic soft tissues. Computer Methods in Applied Mechanics & Engineering, 151, 513538. Argoubi, M. and ShiraziAdl, A. (1996). Poroelastic creep response analysis of a lumbar motion segment in compression. Journal of Biomechanics, 29, 13311339. Armstrong, C. G., Lai, W. M. and Mow, V. C. (1984). An analysis of the unconfined compression of articular cartilage. Journal of Biomechanical Engineering, 106, 165173. Ateshian, G. A. and Wang, H. Q. (1995). A theoretical solution for the frictionless rolling contact of cylindrical biphasic articular cartilage layers. Journal of Biomechanics, 28, 13411355. Bear, J. (1972). Dynamics of Fluids in Porous Media. Elsevier, New York. Bear, J. (1990). Introduction to Modeling and Transport Phenomena in Porous Media. Kluwer, Dordrecht. Bender, C. M. and Orszag, S. A. (1978). Advanced Mathematical Methods for Scientists and Engineers. McGrawHill International, Singapore. B iot, M. A. (1941 a). General theory of threedimensional consolidation. Journal of Applied Physics, 12, 155165. Biot, M. A. (1941b). Consolidation settlement under a rectangular load distribution. Journal of Applied Physics, 12, 426430. Biot, M. A. (1955). Theory of elasticity and consolidation for a porous anisotropic solid. Journal of Applied Physics, 26, 182185. Biot, M. A. (1956a). Theory of propagation of elastic waves in a fluidsaturated porous solid (Parts I and II). Journal of the Acoustical Society of America, 28, 168191. Biot, M. A. (1956b). General solutions of the equations of elasticity and consolidation for a porous material. Journal of Applied Mechanics, 23, 9196. Biot, M. A. (1956c). Thermoelasticity and irreversible thermodynamics. Journal of Applied Physics, 27, 240253. Biot, M. A. (1962). Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics, 33, 14821498. Biot, M. A. (1964). Theory of buckling of a porous slab and its thermoelastic analogy. Journal of Applied Mechanics, 31, 194198. Biot, M. A. and Willis, D. G. (1957). The elastic coefficients of the theory of consolidation. Journal of Applied Mechanics, 24, 594601. Burridge, R. and Keller, J. B. (1981). Poroelasticity equations derived from microstructure. Journal of the Acoustical Society of America, 70, 11401146. Carroll, M. M. (1979). An effective stress strain law for anisotropic elastic deformation. Journal of Geophysical Research, 84, 75107512. Cederbaum, G. (2000a). Postbuckling behavior of poroelastic columns. International Journal of Mechanical Sciences, 42, 771783. Cederbaum, G. (2000b). Dynamic stability of poroelastic columns. Journal of Applied Mechanics, in press. Cederbaum, G., Schulgasser, K. and Li, L. P. (1998). Interesting behavior patterns of poroelastic beams and columns, International Journal of Solids and Structures, 3435, 49314943. Cheng, A. H.D., Abousleiman, Y. and Roegiers, J. C. (1993). Review of some poroelastic effects in rock mechanics. International Journal of Rock Mechanics and Mining Science & Geomechanical Abstracts, 30, 11191126. Cleary, M. P. (1977). Fundamental solutions for a fluidsaturated porous solid. International Journal of Solids and Structures, 13, 785806.
142
References
Coussy, O. (1995). Mechanics of Porous Media. Wiley, Chichester. Cowin, S. C. (1999). Bone poroelasticity. Journal of Biomechanics, 32, 217238. Cryer, C. W. (1963). A comparison of the threedimensional consolidation theories of Biot and Terzaghi. Quarterly Journal of Mechanics and Applied Mathematics, 16, 401412. Detourney, E. and Cheng, A. H.D. (1993). Fundamentals of poroelasticity, in: Comprehensive Rock Engineering: Principles, Practice & Projects, Vol. II (J. A. Hudson, ed.). Pergamon, Oxford. Eisenberg, S. R. and Grodzinsky, A. J. (1985). Swelling of articular cartilage and other connective tissues. Journal of Orthopaedic Research, 31, 48159. Eisenberg, S. R. and Grodzinsky, A. J. (1987). The kinetics of chemically induced nonequilibrium swelling of articular cartilage and corneal stroma. Journal of Biomechanical Engineering, 109, 7989. Goldsmith, A. A. J., Hayes, A. and Clift, S. E. (1996). Application of finite elements to the stress analysis of articular cartilage. Medical Engineering & Physics, 18, 8998. Gu, W. Y., Lai, W. M. and Mow, V. C. (1993). Transport of fluid and ions through a porouspermeable chargedhydrated tissue, and streaming potential data on normal bovine articular cartilage. Journal of Biomechanics, 26, 709723. Gu, W. Y., Lai, W. M. and Mow, V. C. (1998). A Mixture theory for chargedhydrated soft tissues containing multielectrolytes: passive transport and swelling behaviors. Journal of Biomechanical Engineering, 120, 169180. Johnson, M. W., Chakkalakal, D. A., Harper, R. A., Katz, J. L. and Rouhana, S. W. (1982). Fluid flow in bone. Journal of Biomechanics, 11, 881885. Kingsbury, H. B. (1984). Determination of material parameters of poroelastic media, in: Fundamentals of Transport Phenomena in Porous Media (J. Bear and M. Corapcioglu, eds.). Nijhoff, Dordrecht. Kumpel, H. J. (1991). Poroelasticity: parameters reviewed. Geophysics Journal International, 104, 783799. Lai, W. M., Hou, J. S. and Mow, V. C. (1991). A triphasic theory for the swelling and deformation behaviors of articular cartilage. Journal of Biomechanical Engineering, 113, 245258. Laible, J. P., Pilaster, D. S., Krag, M. H., Simon, B. R. and Haugh, L. D. (1993). A poroelasticswelling finite element model with application to the intervertebral disc. Spine, 18, 659670. Lanir, Y. (1987). Biorheology and fluid flux in swelling tissues. II. Analysis of unconfined compressive response of transversely isotropic cartilage disc. Biorheology, 24, 189205. Li, L. P. (1997). Theory of poroelastic structures with diffusion in the longitudinal directions. Ph.D. thesis, BenGurion University of the Negev, Beer Sheva, Israel. Li, L. P., Cederbaum, G. and Schulgasser, K. (1996). Vibration of poroelastic beams with axial diffusion. European Journal of Mechanics: A/Solids, 15, 10771094. Li, L. P., Schulgasser, K. and Cederbaum, G. (1995). Theory of poroelastic beams with axial diffusion. Journal of the Mechanics and Physics of Solids, 43, 20232042. Li, L. P., Schulgasser, K. and Cederbaum, G. (1997a). Buckling of poroelastic columns with axial diffusion. International Journal of Mechanical Sciences, 39, 409415. Li, L. P., Schulgasser, K. and Cederbaum, G. (1997b). Large deflection analysis of poroelastic beams. International Journal of NonLinear Mechanics, 33, 114. Li, L. P., Cederbaum, G. and Schulgasser, K. (1997c). Theory of poroelastic plates with inplane diffusion. International Journal of Solids and Structures, 34, 45154530. Li, L. P., Cederbaum, G. and Schulgasser, K. (1999a). A finite element model for poroelastic beams with axial diffusion. Computers and Structures, 73, 595608. Li, L. P., Soulhat, J., Buschmann, M. D. and ShiraziAdl, A. (1999b). Nonlinear analysis of cartilage in unconfined ramp compression using a fibril reinforced poroelastic model. Clinical Biomechanics, 14, 673682. Li, L. P., Buschmann, M. D. and ShiraziAdl, A. (2000). A fibril reinforced nonhomogeneous poroelastic model for articular cartilage. Journal of Biomechanics, submitted. Mak, A. F. (1986a). Unconfined compression of hydrated viscoelastic tissues: a biphasic poroviscoelastic analysis. Biorheology, 23, 371383. Mak, A. F. (1986b). The apparent viscoelastic behavior of articular cartilage  the contributions from the intrinsic matrix viscoelasticity and interstitial fluid flows. Journal of Biomechanical Engineering, 108, 123130. Mak, A. F., Lai, W. M. and Mow, V. C. (1987). Biphasic indentation of articular cartilage. I. Theoretical analysis. Journal of Biomechanics, 20, 703714. Mandel, J. (1953). Consolidation des sols (etude mathematique). Geotechnique, 3, 287299.
References
143
McNamee, J. and Gibson, R. E. (1960a). Displacement functions and linear transforms applied to diffusion through porous elastic media. Quarterly Journal of Mechanics and Applied Mathematics, 8, 98111. McNamee, J. and Gibson, R. E. (1960b). Plane strain and axially symmetric problems of the consolidation of a semiinfinite clay stratum. Quarterly Journal of Mechanics and Applied Mathematics, 8, 210227. Mow, V. C., Ateshian, G. A. and Spilker, R. L. (1993). Biomechanics of diarthroidal joints  a review of 20 years of progress. Journal of Biomechanical Engineering, 115, 460467. Mow, V. C., Holmes, M. H. and Lai, W. M. (1984). Fluid transport and mechanical properties of articular cartilage: a review. Journal of Biomechanics, 17, 377394. Mow, V. C., Kuei, S. C., Lai, W. M. and Armstrong, C. G. (1980). Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments. Journal of Biomechanical Engineering, 102, 7384. Murad, M. A. and Loula, A. F. D. (1992). Improved accuracy in finite element analysis of Biot's consolidation problem. Computer Methods in Applied Mechanics and Engineering, 95, 359382. Murad, M. A. and Loula, A. F. D. (1994). On stability and convergence of finite element approximations of Biot's consolidation problem. International Journal of Numerical Methods in Engineering, 37, 645667. Nowacki, W. K. (1986). Thermoelasticity, 2nd edn. Pergamon Press, Oxford. Nowinski, J. L. (1971). Bone articulations as systems of poroelastic bodies in contact. AIAA Journal, 9, 6269. Nowinski, J. L. (1972). Stress concentrations around a cylindrical cavity in a bone treated as a poroelastic body. Acta Mechanica, 13, 281292. Nowinski, J. L. and Davis, C. F. (1970). A model of the human skull as a poroelastic spherical shell subjected to a quasistatic load. Mathematical Biosciences, 8, 397416. Nowinski, J. L. and Davis, C. F. (1972). The flexure and torsion of bones viewed as anisotropic poroelastic bodies. International Journal of Engineering Science, 10, 10631079. Prendergast, P. J. (1997). Finite element models in tissue mechanics and orthopaedic implant design. Clinical Biomechanics, 12(6), 343366. Rajapakse, R. K. N. D. (1993). Stress analysis of borehole in poroelastic medium. Journal of Engineering Mechanics, 119, 12051227. Rice, J. R. and Cleary, M. P. (1976). Some basic stress diffusion solutions for fluidsaturated elastic porous media with compressible constituents. Reviews of Geophysics and Space Physics, 14, 227241. Rudnicki, J. W. (1985). Effect of pore fluid diffusion on deformation and failure of rock, in: Mechanics of Geomaterials (Z. Bazant, ed.). WileyInterscience, Chichester. Sandhu, R. S. and Wilson, E. L. (1969). Finiteelement analysis of seepage in elastic media. Journal of Engineering Mechanics, 95, 641651. Schulgasser, K. and Witztum, A. (1992). On the strength, stiffness and stability of tubular plant stems and leaves. Journal of Theoretical Biology, 155, 497515. Setton, L. A., Zhu, W. B. and Mow, V. C. (1993). The biphasic poroviscoelastic behavior of articular cartilage role of the surface zone in governing the compressive behavior. Journal of Biomechanics, 26, 581592. Simon, B. R. (1992). Multiphase poroelastic finite element models for soft tissue structures. Applied Mechanics Reviews, 45, 191218. Simon, B. R., Wu, J. S., Carlton, M. W., Kazarian, L. E., France, E. P., Evans, J. H. and Zienkiewicz, O. C. (1985a). Poroelastic dynamic structural models of rhesus spinal motion segments. Spine, 10, 494507. Simon, B. R., Wu, J. S., Carlton, M. W., Evans, J. H. and Kazarian, L. E. (1985b). Structural models for human spinal motion segments based on a poroelastic view of the intervertebral disk. Journal of Biomechanical Engineering, 107, 327335. Spilker, R. L. and Suh, J. K. (1990). Formulation and evaluation of a finite element model for the biphasic model of hydrated soft tissues. Computers and Structures, 35, 425439. Spilker, R. L., Donzelli, P. S. and Mow, V. C. (1992a). A transversely isotropic biphasic finite element model of the meniscus. Journal of Biomechanics, 25, 10271045. Spilker, R. L., Suh, J. K. and Mow, V. C. (1992b). A finite element analysis of the indentation stressrelaxation response of linear biphasic articular cartilage. Journal of Biomechanical Engineering, 114, 191201. Stevens, K. K. (1966). On the parametric excitation of a viscoelastic column. AIAA Journal, 12, 21112116. Suh, J. K. and Bai, S. (1998). Finite element formulation of biphasic poroviscoelastic model for articular cartilage. Journal of Biomechanical Engineering, 120, 195201. Suh, J. K. and DiSilvestro, M. R. (1999). Biphasic poroviscoelastic behavior of hydrated biological soft tissue. Journal of Applied Mechanics, 66, 528535.
144
References
Taber, L. A. (1992). A theory for transverse deflection of poroelastic plates. Journal of Applied Mechanics, 59, 628634. Terzaghi, K. (1943). Theoretical Soil Mechanics. Wiley, New York. Theodorakopoulos, D. D. and Beskos, D. E. (1993). Flexural vibrations of fissured poroelastic plates. Archive of Applied Mechanics, 63, 413423. Theodorakopoulos, D. D. and Beskos, D. E. (1994). Flexural vibrations of poroelastic plates. Acta Mechanica, 103, 191203. Thompson, M. and Willis, J. R. (1991). A reformation of the equations of anisotropic poroelasticity. Journal of Applied Mechanics, 58, 612616. Touati, D. and Cederbaum, G. (1994). Dynamic stability of nonlinear viscoelastic plates. International Journal of Solids and Structures, 31, 23672376. Vankan, W. J., Huyghe, J. M., Janssen, J. D. and Huson, A. (1996). Poroelasticity of saturated solids with an application to blood perfusion. International Journal of Engineering Science, 34, 10191031. Vankan, W. J., Huyghe, J. M., Janssen, J. D., Huson, A., Hacking, W. J. G. and Schreiner, W. (1997). Finite element analysis of blood flow through biological tissue. International Journal of Engineering Science, 35, 375385. Washizu, K. (1982). Variational Methods in Elasticity & Plasticity, 3rd edn. Pergamon Press, New York. Weier, T. E., Stocking, C. R., Barbour, M. G. and Rost, T. L. (1982). Botany: an Introduction to Plant Biology (Chapter 12), 6th edn. Wiley, New York. Wu, J. Z., Herzog, W. and Ronsky, J. (1996). Modeling axisymmetrical joint contact with biphasic cartilage layers  an asymptotic solution. Journal of Biomechanics, 29, 12631281. Zhang, D. and Cowin, S. C. (1994). Oscillatory bending of a poroelastic beam. Journal of the Mechanics and Physics of Solids, 42, 15751599.
Appendix A PROOF OF THE VARIATIONAL PRINCIPLES
Noting Eqs. (2.14), (4.3) and (2.28) and the fact that a variation of an initial quantity equals zero, the variation of U is as follows: 6U
I
[ (2EAu'x + riNP)* 6(Ux) ' + riit'x * 6Np
AEA rI Np(x,O)6Np
+ (2EIw,xx  riMp), 6(ft,xx)  rift,~x * 6Mp + AEI rI Mp(x, 0)6Mp ]dx
(A.1)
It has been assumed that the order of a variation and a derivative can be interchanged. After integration by parts of the first and fourth terms once and twice respectively, (A.1) becomes 68
l[ L
(2EAu,xx + riNp,x)* t~ti + ritix * 6Np +  ~
+ (2EIw,xxxx 
riMp,xx)* ~ft  rlft:,xx * 6Mp +
Np(x, O),~Np
ri Mp(x, O)(~Mp]dx AEI
+ [(~A.~ + ~,~. ~]~+[.~,Wxx.~. ~Wx~]~ [(2EIw,xxx
riMp,x)* t~ft]S
(A.2)
Since the variations of the loads and the given boundary quantities are zero, the variation of V is given as 6V  2 f [qn * 6ft  qs * 6u]dx  [2N * 6ti] b + [2/17/* 6(ftx)] b ./ L
[
Up *
,

  ~ Mp * 6(Mp,x) a
Consider now the variation of the functional (4.6)
(A.3)
Appendix A
146
t~TrmP AEIT] f [ _A l (KNpxx ' _ lVp + AEAit,x) , t3Np + (KMpxx ' _ Mp _ AEifv 'xx) , I
+ ~ Np * 6(KNp,xx  Np ~ AEAu,x) + Mp * 6(KMp,xx  Mp  ~EIw,xx)]dx (A.4) After integrating by parts or rewritten by (4.3) when necessary, it becomes 671"mP AEI rl I
[ A I (2KNpxx '  21~Ip + AEAitx) ' * 6Np  AEINp'x * 6it  ~ Np(x, O)6Np
+(2XMp,xx  2Mp  AEIwxx) * 6Mp  AEIMp,xx * 6w  Mp(x, 0)6Mp ]dx
[
][
nt[T~Xp* ~/i]~[  ~ X p * 6(Xp,x)a ~~
a
Adding (A.2), (A.3) and (A.5) together, the variation of the functional 7r = U + V + is obtained as follows:
6"n' 2 f [(EAu,xx .I L k
+ ~Np,x +
qs)* 6u + (EIw,xxxx  ~Mp,xx +
+ AEArl (KNp~x _ IWp q AEAft,x) * 6Np +  ~ (KMp,xx 
7rmp
qn)*W
Mp 
hEIw,xx)* 6Mp ]dx
+[2(N  / V ) * &i]ab [2(M  19/)* 6(W,x)lab + [2(Ma  ~))* 6wlab + [ ~r/g
(Np /Vp) * t~(Np,x) ]i + [ ~K(MpMp)*a(Mpx)]ba ~ , (A.6)
If a displacement is given at a boundary, the variation of the displacement at the boundary is zero, thus the corresponding item vanishes. Noting that every item in (A.6) is independent and every nonzero variation is arbitrary, it is seen that 67r  0 gives the equilibrium equations (2.16), the motion equations (2.22), the mechanical boundary conditions (2.24) and the diffusion boundary conditions (2.25) and/or (2.26). In other words, the stationary condition of 7r is identical to those governing equations and all the boundary conditions except for the displacement boundary conditions (2.23) which are preconditions. Thus it has been proved that 7r corresponds to a
Appendix A
147
variational principle which includes two types of variables, the displacements and the pore pressure resultants, and hence refers to a generalized variational principle. Similarly, for the second functional 7r*, the following can be obtained: 6"n'*  Jf I_[(EAuxx + rINp'x + qs)* 6u + (Elwxxxx  rIMp,xx + qn)* 8w L
+
'
rl * (KNpxx  IVp k hEAit,x ) * 6Np + rl * AEA ' AEI
(KMp,xx l~p
 l~Ell,V,xx) * ~Mp ]dx
+ [(N /~') * 6u]~[(M/17/), 8(W,x)]~+ [(M,x  {~)* 6W]ba
+

2AEA * (Np /Vp) * ~(Np,x)
2AEA
* Np,~ * 6Np

2AEI
+ 2AEI 9(Mp  Mp) 9 6(Mp,x) * Mp,x * 8Mp
(A.7)
by which we see that the stationary condition of zr* gives the same EulerLagrange equations.
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Appendix
A FINITE ELEMENT
FOR POROELASTIC
B
BEAMS
A 3node element is given here as an example of the beam finite elements, which are formulated earlier. The nodes are numbered as 1, 2 and 3. At each node, all the four unknown variables are adopted as nodal parameters. The nodal unknowns of the element are recorded as
{ T w }  [Wl, 01, W2, 02, W3' 03] T
(B. 1)
{/~M }
_
[ (Mp)l, (Mp)2, (Mp)3
]~
where Oi is the derivative of w with respect to x at the node i (  1,2,3). For convenience, we define a polynomial as
Pi(x)  1 7 (x  xj) j#i
Then the interpolation matrices are given as
[Nu] = [SN] = [SM] [Pl(x)/Pl(Xl),
P2(x)/P2(x2),
Pa(x)/P3(x3)]
(S.2)
which is based on the Lagrange interpolation and
[Nw] = [al, bl, a2, b2, a3, b3]
(B.3)
where a i and bi, according to the Hermite interpolation, are as follows:
a i 
[
2 1

Pi(xi)
Pi(xi)
,
bi 
(x 
xi)
Pi(xi)
in which the derivative of the polynomial at the node i is
Pli(xi)  3 x i  (x 1 nt x 2 nt x 3) It should be noticed that the interpolation matrices [N,], [SN] and [SM] are independent quantities although they may have the same forms when the same nodes are adopted for u, Up and Mp.
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Appendix C SEVERAL USEFUL LAPLACE INVERSE TRANSFORMATIONS
In order to find the inverse transform of Eq. (5.10) for all cases so that the Ogn(t) in (5.7) are determined, some formulas for Laplace inverse transformations are required. The following two groups of formulas are derived for this purpose, using ~ to refer to the Laplace operator,
~,1{ s ~,_1I 9o1{
~L~,~(t)]l _ eat ~tof(~.)e_a~ d~., 1
( S  a) 2
~[f(t)]}eatftof(~')ea~(t~)
1 ~ [ f ( t ) ] } (s _ a)3
eat f('r)ear(t ~f'o
d~,
(C.la, b, c)
T)2dT
and
~1{
(s  sce a~2 ~_/32 ~[b(t)]
}

e c~t[cos(flt)Ic(t) + sin(fit,Is(t)] (C.2a, b)
(s  a) 2 +/32 5~

[sin(flt)Ic(t)  cos(fit)Is(t)]
where
Ic(t) 
it b(z)e~'cos(flz)dz, o
I~(t) 
;t b(z)e~'sin(flz)dr o
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Appendix D COEFFICIENTS IN DIFFERENCE FORMULAS
The coefficients for all difference formulas required in the finite difference method for our problem are presented in this appendix. All of them are derived by Lagrange interpolation and for the computation of derivatives at the node i. The coefficients for the 3node difference formulas are given in Table D. 1, in which the first to third rows present the coefficients of Eqs. (6.28a,b,c), respectively. For second differences, only the central formula is employed in our work; the coefficients for the 3node formula are given in the fourth row of Table D.1. T a b l e D.1. Coefficients for 3node formulas ( h i   x i  X i  1 ) F o r w a r d formula
Central formula
 2 h i + 1  hi+2 + hi+z)
C~ =
Co = h i + l ( h i + l
Ci
i C~ =
9 hi Ct2  h i _ l ( h i _ 1 4 hi)
2rid order (central)
C/
C~ =
hi+l
(hi+l + hi+2)hi+ 2
hi+lhi+2
_ hi+l 1  hi(h i + hi +1)
Backward formula
hi+ 1 k hi+ 2
Ci
2
+ hi+l hihi+ 1
_  h i  1  hi 1 hi_lhi 9
1  h i ( h i + hi+ 1)
hi
2
i hi C1 = (hi + hi+l)hi+l i h i  1 + 2hi C~ = hi(hi_ ~ ~hi) i 2 C1  (hi + hi+ 1)hi+l
Cl~  hihi+ 1
The 5node formulas are employed in the work for the node i which is at least two nodes away from an end node so that the central formula can be applied. The coefficients for the first order difference as defined in Eq. (6.29) are presented in Table D.2. Table D.2. Coefficients for 5node central f o r m u l a (k  i  2 to i + 2)
c'_~
c~_,
II kr
(Xi  Xk) k#i
H (Xi2  Xk) k#i2
II kr162
c~ (Xi  Xk)
II (Xi1  Xk) k#i 1
~,
c~
c~ 1
kZ'~r Xi  Xk
II kr162
(Xi  Xk)
II (Xi+I  Xk)
k#i+ 1
II k#i,kr
H
k#i+2
(Xi  Xk)
(Xi+2  Xk)
The second order difference formula is written in the form of Eq. (6.35). The coefficients for the central formula are as follows:
Appendix D
154
CL 2 =
2[(Xi  X i _ l ) ( X
i 
Xi+l) +
(X i  Xi_l)(X i 
kr
CL1 =
2[(Xi  X i _ 2 ) ( X
i 
Xi+l) +
I'I (Xi2  2
 Xi+l)(X i 
Xi+2) ]
(X i   X i + l ) ( X i  
Xi+2) ]
 xk)
(X i  Xi_2)(X i 
11
Xi+2) + (X i
Xi+2) +
(xilxk)
kr
1
e 0l
2[(Xi  X i _ 2 ) ( X
i  Xi_l)
+ ( X i   X i + l ) ( X i  
Xi+2) qt_ ( 2 X i
_ Xi_2 _ Xi_l)(2X
i _ Xi+l
_
Xi+2)]
[ 1 (xi  x O kr
C~= 2[(xi

xi_2)(x
i 
Xi_l)
+ (x i 
xi_2)(x
H kr
i 
xi+2) +
(x i 
Xi_l)(X i 
xi+2) ]
(xi+ 1  Xk) + 1
c~ = 2 [ ( x /  xi2)(xi  xi~) + (xi  xi2)(xi  xi+~) + (xi  xi~)(xi  xi+ ~)] l'I kr
(Xi+2 
xk)
+ 2
(D.1)
Appendix E D E T E R M I N A T I O N OF B O U N D A R Y VALUES AT xl FOR THE FINITE D I F F E R E N C E M E T H O D
The fact that the boundary conditions for w, 0, M and Q produce coupling in the variables requires that the four unknowns be solved dependently. In the solution scheme presented in this book, we avoid solving them simultaneously due to considerations of convergence. Instead, the numerical solutions are pursued first on the supposition that all of the four variables, no matter whether they are previously given or not, are zero at x = X1. All of the four nodal parameters at Xl (i.e. wl, 01, M1 and Q1, or their variations, or their increments) can then be determined by four given boundary conditions and the supposed solutions. The real solutions are obtained by modifying the supposed solutions with the four nodal parameters. The procedures are presented in this appendix. We derive necessary equations for time t = 0 only. However, as will be seen later, they can be easily adapted to all times. Consider first the corresponding linear case, since its solution is adopted for the first loading step in an iterative approach when very small loading values (include the given boundary values) are taken. The governing equations for the four unknowns at t = 0, in accordance with Eqs. (6.30), are as follows: dw dx  0 '
dO dx  
M I+A~'
dM dx  Q'
dQ dx  q n
(E.1)
Supposing that wl  0, 01  0, M1 = 0 and Q1 = 0, we get a supposed solution for (E. 1) recorded as Wsup, 0sup, msup and Qsup. The relationship between the real solution and the supposed solution can be analytically determined by (E.1). Integrating each equation of (E.1) from the last to the first successively yields Q0.
c1
c3
~ Qsup,
M
c 2 Jr C l X Jr
2c2x + Clx2 .2(1 + At/) . . + 0sup'
w
Msup, c4
+ c3x
3c2x2 + Clx3 6(1 + At/) + Wsup
(E.2)
from which four linear algebraic equations can be obtained in any case by the four given boundary conditions and thus the constants c l, c2, c3 and c4, whose values in fact are the real values of Q1, M1, 01 and Wl, respectively, will be determined. Hence, (E.2) gives the
156
Appendix E
real solution. In practice, we solve for the four constants using the supposed numerical solution. In a nonlinear situation, when the boundary conditions cannot be decoupled, Eqs. (6.30) are replaced by their variational form (6.32). Similarly, the real solution of the variations is given by o~O = c 1 + (o~Q)sup 6kM
= c 2 4 r x 4 (o~'kM)sup
(E.3)
2c2 x 4 c1 x2
0 = c3 ~'kW  C4 4
2(1 + A~) + ( o~ 0)sup
0
C3 
2(1 + At/)
(COs0)kldx 4 (~'kW)sup
The algebraic equations for the constants given by the four boundary conditions are also linear though integral relationships exist among the coefficients, which make numerical calculations necessary. Since ~Q, o~M, o~0 and o~w are obtained by iterations for any given k, the above procedure must be applied to every iteration. Apparently, the same can be done for time t > 0. Eqs. (E.2) and (E.3) are justified for the situation after (1 + At/) is removed from the two equations and o~ in (E.3) is replaced by AJ in accordance with Eqs. (6.34). For instance, the latter should be as follows: AJQ = cl + (AJQ)sup NM
  c 2 4 c1 x 4
zxJ0 _ c3 _
AJw
c4 4
(AJM)sup
2c2 x 4 c1 x2
2
o
c3 
(E.4)
4 (N0)su p
2
(COS0r
dx 4 (~Jw)sup
SUBJECT INDEX anisotropic, 3 antisymmetric, 24, 25, 45, 85, 127 beam
cantilever, 2832, 4648, 75, 79, 84, 87, 90 clamped, 4748, 85 drained, 14, 15, 16,20,21,26,59, 61, 62,91,107, 110 simply supported, 15, 16, 2127, 41, 45, 54, 59, 65, 72, 85, 89 sealed, 16, 18, 46, 50, 79, 80, 83, 91 statically determinate, 14, 20, 89 biomechanics, 1, 2, 7 Biot, 1, 36, 10, 13, 33, 67, 91, 96, 111, 115 bone, 2 boundary conditions, 2, 4, 5, 1314, 18, 26, 34, 4251, 146, 155 buckling, 3, 6, 8996 bulk, 2, 4, 12, 67, 111 cartilage, 2 characteristic times, 1516, 59, 61, 134, 136 column, 6, 4952 consolidation, 1 constitutive law, 5, 6, 11, 17, 111, 138 control, 7, 66, 135, 136 convergence, 41 convolution, 5, 3334, 40 coupling, 1 creep, 47, 79 cylindrical bending, 124128 damping critical, 5, 55, 59, 65 light, 5, 6, 59, 60 over, 5, 59 structural, 66 Darcy's law, 46, 12, 33, 67, 70, 96, 97, 111, 115, 139 deflection large, 5, 6, 6787, 96 small, 46, 10, 67, 70, 71, 74, 75, 79, 80, 84, 87, 89, 99 diffusion boundary condition, 46, 7, 14, 24, 27, 35, 40, 4345, 66, 87, 89, 116, 117
thermal, 1 diffusion time, 134 discrete, 34, 3940, 136139 discretization, 7577 dissipation time 59, 61 drained, 6, 14, 15, 39 end clamped, 47,48 free, 4344, 4748 impermeable, 14, 16, 22, 41, 44, 4749 permeable, 14, 21, 47, 53, 63, 65 equilibrium, 1, 4, 5, 9, 12, 18, 39, 67, 69, 72, 114, 117, 138, 146 EulerLagrange equations, 5, 15, 34, 36 finite difference method, 5, 7279, 155 finite element method, 3341 fluid compressibility, 11, 12, 114 Fourier series, 19, 54, 63, 125, 128 frequency, 15, 6162, 65, 108, 131134 functional, 5, 3337, 41 geomechanics, 1 Galerkin method, 33 imperfection sensitivity, 6, 89, 102104 impermeable, 14, 16, 21, 22, 23, 77, 116 inertia, 4, 9, 10, 35, 106, 119, 138 infinite time, 34, 41, 80, 94 initial conditions, 4, 13, 14, 16, 18, 21, 29, 53, 60, 62, 70, 117 instantaneous response, 14, 25, 79, 84 interpolation, 5, 37, 39, 41, 73, 149, 153 jacketed, 12 Kirchoff hypothesis, 6, 111 Kirchoff plate 3, 112 kinematic, 10 Lagrange multipliers, 5, 3336 Laplace transform, 5, 54, 56, 151 loads critical, 6, 9095, 104, 107110 suddenly applied, 12, 1416, 65, 90
Subject index
158 uniformly distributed, 79, 85, 87 MandelCryer effect, 7, 29, 4244, 48 mass, 9, 10 Mathieu equation, 107 microgeometry, 3, 9, 114 modulus bulk, 2 drained, 114 Young's, 12, 14, 19, 20, 21, 65, 114, 117 multiple scale method, 107108 nondimensional, 9, 16, 1718, 19, 53, 61, 71, 72, 97, 118, 138 nonlinear, 5, 67, 72, 74, 75, 77, 96, 98, 101,156 orthopedic, 2 orthotropic, 6, 111, 115 overshooting, 26, 29 permeable, 14, 16, 20, 22, 53, 63, 65 plate, 24, 6, 42, 111134 rectangular, 120134 square, 122, 124, 131132 Poisson's ratio, 114, 117, 138140 pore pressure, 5961, 65 pore volume, 9, 11, 12 porosity 9, 11 postbuckling, 6, 96104 potential energy, 35, 131 quasistatic, 1, 4, 5, 6, 19, 34, 65, 89, 105, 119, 132 relaxation, 47 resonance, 5, 7, 6466 roots
complex, 55, 108, 129 conjugate, 57, 108, 129 real, 55, 56 RungeKutta method, 99 saturated, 1, 2, 6, 13, 66, 89, 107 secular terms, 108109 separation of variable method, 21 skeletal material, 3, 5, 11, 50, 135 smart structures, 7, 65, 135 soft tissue, 2 stability boundaries 6, 89, 107110 condition, 6, 89 dynamic, 6, 89, 104110 steady state, 127 stem, 3 Strutt diagram, 110 superposition, 39 symmetric, 9 thermoelastic analogy, 1 transversely isotropic, 3, 4, 9, 10, 11, 67, 111, 113 trapped, 12, 15, 16, 62, 117, 138 variational principle, 5, 3437 vibrations, 5366, 128133 forced, 5, 6365, 132134 free, 5, 15, 5963, 65, 129131 viscoelastic, 67, 24, 26, 46, 47, 66, 110, 135 viscosity, 7, 13, 66, 70, 90 volumetric change, 1 water, 3, 66 wave propagation, 1